We advance here an algorithm of the synthesis of lossless electric circuits such that their evolution matrices have the prescribed Jordan canonical forms subject to natural constraints. Every synthesized circuit consists of a chain-like sequence of LC-loops coupled by gyrators. All involved capacitances, inductances, and gyrator resistances are either positive or negative with values determined by explicit formulas. A circuit must have at least one negative capacitance or inductance for having a nontrivial Jordan block for the relevant matrix.

This work is motivated by an interest in electromagnetic and optical systems exhibiting Jordan eigenvector degeneracy, which is a degeneracy of the system evolution matrix when not only some eigenvalues coincide but the corresponding eigenvectors coincide also. Another way to describe the eigenvector degeneracy of a matrix is by acknowledging that there is no basis in the relevant vector space made of eigenvectors of the matrix. Such degenerate system states are quite often referred to as exceptional points of degeneracy (EPDs) (Ref. 1, II.1). A particularly important class of applications of EPDs is sensing.2–5 Other potential applications include (i) enhancement of the gain in active systems6–10 and (ii) directivity of antennas.11 A variety of systems have been suggested that exhibit EPDs in space for waveguide structures and time for circuits. These systems are based on (i) non-Hermitian parity-time (PT) symmetric coupled systems, which are systems with balanced loss and gain;12–14 (ii) coupled resonators;15–17 and (iii) electronic circuits involving dissipation.18 

Systems with EPDs in the literature cited above commonly involve loss and gain elements, suggesting that they might be essential to the existence of EPDs (see, for instance, Ref. 19). It turns out, though, that the presence of loss and gain elements in a system is not necessary for having EPD regimes. An interesting system without loss and gain elements has been proposed in Ref. 20 where the authors demonstrate that EPDs can exist for a single LC resonator with time-periodic modulation. Our own studies in Ref. 21 show that an analytical model of a traveling wave tube (TWT) has the Jordan eigenvector degeneracy at some points of the system dispersion relation. This TWT system is governed by a Lagrangian, and consequently, it is a perfectly conservative system. Inspired by those studies, we raised a question if simple lossless (perfectly conservative) circuits exist such that their evolution matrices exhibit the Jordan eigenvector degeneracy. We answered to the question positively by constructing circuits with prescribed degeneracies.

Our primary goal here is to synthesize a lossless electric circuit so that its evolution matrix H has a prescribed Jordan canonical form J subject to natural constraints considered later on. Hence, by the definition of the Jordan canonical form, H=SJS1, where S is an invertible matrix and J is a block diagonal matrix of the form

J=Jn1ζ1000Jnq1ζq1000Jnqζq,Jnζ=ζ1000ζ1000ζ1000ζ.
(1.1)

ζj are real or complex numbers, and Jnζ is the so-called Jordan block, which is an n × n matrix. For n = 1, the matrix Jnζ=ζ turns just into number ζ.

As to the evolution matrix H, we assume that the circuit evolution is governed by the following linear equation:

tX=HX,
(1.2)

where X is the 2n dimensional vector-column describing the circuit state and H is the 2n × 2n matrix, where n > 1 is an integer. The particular choice of the dimensions is explained by our desire to have an underlying Lagrangian and Hamiltonian structure so that Eq. (1.2) will be the Hamilton evolution equation. Consequently, the 2n × 2n matrix H is going to be a Hamiltonian matrix, and we will refer to it as the circuit evolution matrix or just circuit matrix (see Sec. VIII). To meet the dimension requirements of the evolution equation (1.2), the circuit topological structure is expected to have nfundamental loops or f-loops for short (see Sec. IX). The circuit state is then described by the corresponding n time-dependent charges qkt, which are the time integrals of the relevant loop currents tqkt. Hamiltonian formulations of the dynamics of LC-circuits have been studied (see, for instance, Ref. 22 and references therein).

The eigenvalue problem associated with the evolution evolution (1.2) is

HX=sX,s=iω,
(1.3)

where ω is the frequency. Note that the eigenvalue (spectral parameter) s is pure imaginary for real frequencies.

As to the prescribed Jordan canonical form J, we are rather interested in the simplest possible systems exhibiting nontrivial Jordan blocks than systems that can have an arbitrary Jordan canonical form allowed for Hamiltonian matrices. It turns out that if the Jordan canonical form J of the circuit matrix H has a nontrivial Jordan block, then the circuit must have at least one negative capacitance or inductance (see Sec. VIII E). The Jordan forms associated with Hamiltonian matrices must satisfy certain constraints considered in Sec. V. The origin of the constraints is the fundamental property of a Hamiltonian matrix H to be similar to HT, which is the transposed to H matrix. This special property of a Hamiltonian matrix combined with the general statement that every square matrix M is similar to the transposed to it matrix MT imposes the following constraints on the spectral structure of matrix H: (i) if s in an eigenvalue of H, then −s is its eigenvalue as well; (ii) the Jordan blocks corresponding to the eigenvalues s and −s have the same structure. If in addition to that the entries of the Hamiltonian matrix H are real-valued, then the following properties hold: (i) if s in an eigenvalue of H, then −s, s̄, and s̄, where s̄ is complex-conjugate to s, are its eigenvalues as well; (ii) the Jordan blocks corresponding to s, −s, s̄, and s̄ have the same structure. We refer to the listed properties as Hamiltonian spectral symmetry (see Secs. V and VIII F). Apart from the Hamiltonian spectral symmetry, the Jordan structure of Hamiltonian matrices can be arbitrary (Ref. 23, 2.2). Our approach to the generation of the Hamiltonian and the corresponding Hamiltonian matrices is intimately related to the Hamiltonian canonical forms (see  Appendix F and references therein).

Another significant mathematical input to the synthesis of the simplest possible systems exhibiting nontrivial Jordan blocks comes from the property of a square matrix M to be cyclic (also called non-derogatory) (see  Appendix B and references therein). We remind that a square matrix M is called cyclic (or non-derogatory) if the geometric multiplicity of each of its eigenvalues is exactly 1 or, in other words, if every eigenvalue of M has exactly one eigenvector. Consequently, if a square matrix M is cyclic, its Jordan form JM is completely determined by its characteristic polynomial χs=detsIM, where I is the identity matrix of the relevant dimension. Namely, every eigenvalue s0 of M of multiplicity m is associated with the single Jordan block Jms0 in the Jordan form JM of M. Consequently, for a cyclic matrixM, its characteristic polynomialχs=detsIMencodes all the information about its Jordan formJM. Another property of any cyclic matrix M associated with the monic polynomial χ is that it is similar to the so-called companion matrix Cχ defined by the simple explicit expression involving the coefficients of the polynomial χ (see  Appendix B and references therein). The companion matrix Cχ is naturally related to the high-order differential equation χtxt=0, where xt is a complex-valued function of t (see  Appendixes B and  D). This fact underlines the relevance of the cyclicity property to the evolution of simpler systems described by higher order differential equations for a scalar function. In light of the above discussion, we focus on cyclic Hamiltonian matrices H for they lead to the simplest circuits with the evolution matrices H having nontrivial Jordan forms J.

Suppose the prescribed Jordan form J is a 2n × 2n matrix subject to the Hamiltonian spectral symmetry and the cyclicity conditions. The synthesis of a circuit associated with J involves the following steps. We introduce first the characteristic polynomial χs=detsI2nJ, which is an even monic polynomial χs of degree 2n. We then consider the companion to χs matrix C (see  Appendix B), which by the design has J as its Jordan form, that is,

C=YJY1,
(1.4)

where the columns of matrix Y form the so-called Jordan basis of the companion matrix C associated with the characteristic polynomial χs=detsI2nJ (see  Appendix B). We proceed with an introduction of our principal Hamiltonian H (see Sec. V) and recover from it the 2n × 2n Hamiltonian matrix H that governs the system evolution according to Eq. (1.2). As the result of our particular choice of the Hamiltonian H, the corresponding to it Hamiltonian matrix H is similar to the companion matrix C, and consequently, it has exactly the same Jordan form J as C. In particular, we construct a 2n × 2n matrix T such that

C=T1HT,H=ZJZ1,Z=TY,
(1.5)

where the columns of matrix Z form a Jordan basis of the evolution matrix H. Relations (1.4) and (1.5) between involved matrices are considered in Sec. VI.

To relate the constructed Hamiltonian H to a circuit, we introduce the corresponding to it Lagrangian L. Finally, based on the Lagrangian L, we design the relevant to it circuit (see Sec. II). Consequently, this circuit evolution is governed by Eq. (1.2) with the cyclic Hamiltonian matrix H that has the prescribedJ as its Jordan form. Each of the described steps of the circuit synthesis and the quantities constructed in the process provide insights into the circuit features.

In the light of our studies, we can revisit now the question whether the presence of the balanced loss and gain is essential for achieving an electric circuit governed by the evolution matrix with nontrivial Jordan forms. We have succeeded in constructing lossless circuits associated with nontrivial Jordan forms. Each of these circuits though must involve at least one negative capacitance or inductance. If we take a look at the physical implementations of negative capacitance and inductance provided in Sec. IX B, we find that they involve matched positive and negative resistances. Based on this, we may conclude that (i) the presence of the balanced loss and gain is essential for achieving negative values for the capacitance and the inductance and (ii) the presence of at least one capacitor or inductor of the negative value of the capacitance or inductance, respectively, is necessary for achieving a lossless electric circuit associated with the nontrivial Jordan form.

Our studies here focus on ideal circuits exhibiting EPDs and the corresponding nontrivial Jordan blocks for the relevant evolution matrices. As to the circuits’ physical implementations, one expects departures from ideally prescribed conditions. For instance, there are natural variations of the values of circuit elements from their nominal values. Another issue is that operational amplifiers that provide for negative capacitances and inductances have their limitations and departures from ideal conditions considered in Sec. IX B. A systematic way to study the effects of deviations from ideally prescribed conditions is to develop constructive perturbation theory at points of degeneracy for advanced here circuits. In fact, we have constructed such a theory and intend to publish it shortly.

The structure of this paper is as follows: In Sec. II, we show our principal circuit tailored to the desired Jordan form J subject to natural constraints. In Sec. III, we introduce special circuits tailored to specially chosen characteristic polynomials χs and the corresponding Jordan forms made of exactly two Jordan blocks of size 2, 3, and 4. To provide maximum flexibility in adjusting the circuit properties for achieving an EPD, we study in Sec. IV the most general case of our simplest circuit composed of two LC-loops coupled by a gyrator, as shown in Fig. 2. We derive there the most general conditions under which the relevant evolution matrix exhibits nontrivial Jordan blocks. In Sec. V, we provide our strategy for the synthesis of circuits associated with the desired Jordan forms. Sec. VI is devoted to the analysis of our principal circuit Hamiltonian, which is the basis to the circuit synthesis. In Sec. VII, we consider the examples of the principal circuit Hamiltonian and significant matrices. Sec. VIII provides aspects of the Lagrangian and Hamiltonian formalisms as well important properties of Hamiltonian matrices. In Sec. IX, we review the basic elements of the electric networks and their elements including gyrators and negative capacitances and inductances.  Appendixes A–F are devoted to a number of mathematical subjects needed for our analysis. In  Appendix G, we provide the list of notations used throughout the paper.

Leaving the technical details of the circuit synthesis to Secs. III-VIII, we present here our principal circuit design that implements the desired Jordan form J of the circuit evolution matrix H. Quite remarkably, the topology of circuits associated with different Jordan forms is essentially the same. The difference between the circuits is in (i) the number of involved LC-loops; and (ii) particular values of the involved capacitances, inductances, and gyration resistances. Figure 1 shows our principal circuit made of n LC-loops coupled by gyrators. Quantities Lj, Cj, and Gj are, respectively, inductances, capacitances, and gyrator resistances.

To simplify equations throughout this paper, we introduce the following dimensionless version of some of the involved quantities:

t̃=ω0t,C̆j=CjCn,L̆j=ω02CnLj,Ğj=GjCnω0,L̆=CnL,
(2.1)

where ω0 > 0 is a unit of frequency and 1 ≤ jn and L̆ is the scaled Lagrangian. To have less cluttered formulas, we actually omit “hat” from L̆, t̃, C̆j, L̆j, and Ğj and simply remember from now on that we use the relevant letters for the dimensionless quantities and the scaled Lagrangian.

FIG. 1.

The principal circuit made of n LC-loops. Note the difference between the left and the right connections for the gyrators and LC-loops. It is explained by the non-reciprocity of the gyrators and is designed to be consistent with (i) the standard port assignment and selection of positive directions for the loop currents and the gyrator and (ii) the sign of the gyration resistance, as shown in Fig. 7 and Eq. (9.2). The values of quantities Lj, Cj, and Gj are determined by Eqs. (2.7) and (2.8), relating them to the coefficients of the relevant polynomial.

FIG. 1.

The principal circuit made of n LC-loops. Note the difference between the left and the right connections for the gyrators and LC-loops. It is explained by the non-reciprocity of the gyrators and is designed to be consistent with (i) the standard port assignment and selection of positive directions for the loop currents and the gyrator and (ii) the sign of the gyration resistance, as shown in Fig. 7 and Eq. (9.2). The values of quantities Lj, Cj, and Gj are determined by Eqs. (2.7) and (2.8), relating them to the coefficients of the relevant polynomial.

Close modal

The principal circuit Lagrangian associated with the principal circuit depicted in Fig. 1 is

L=k=1nLktqk22k=1nqk22Ck+k=1n1Gkqktqk+1qk+1tqk2,qk=qkt=ikdt,1kn,
(2.2)

where qk and ik are, respectively, the charges and the currents associated with LC-loops of the principal circuit depicted in Fig. 1. The corresponding Euler–Lagrange (EL) equations are

L1t2q1G1tq2+q1C1=0,Lnt2qn+Gn1tqn1+qnCn=0,
(2.3)
Lkt2qk+Gk1tqk1Gktqk+1+qkCk=0,1<k<n.
(2.4)

It is well known that the EL equations (2.3) and (2.4) represent the Kirchhoff voltage law for each of the n f-loops (see Sec. IX). Indeed, each term in these equations is associated with the voltage drop for the relevant electric element as it can be verified by comparison with the voltage–current relations reviewed in Sec. IX A. As to the Kirchhoff current law, one finds that it is already enforced by the selection of n involved f-loops and currents tqk there. Indeed, according to the gyrator settings, the kth gyrator has exactly two incoming currents tqk and tqk+1. Indeed, the outgoing current, which passes through the gyrator branch common to the kth and the k+1-th f-loops, is equal to the sum tqk + tqk+1. When exiting this common branch, the current tqk + tqk+1 splits into currents tqk and tqk+1 in perfect compliance with the Kirchhoff current law. Note that each of the two equations in (2.3) corresponds to the first and the last f-loops and involves only a single gyration resistance. Each of the other f-loops has two adjacent f-loops, and consequently, the relevant to it equation in (2.4) involves two gyration resistances. Note also the difference between left and right connections between the gyrators and LC-loops. It is explained by the non-reciprocity of the gyrators and is designed to be consistent with (i) the standard port assignment and selection of positive directions for the loop currents and the gyrator and (ii) the sign of gyration resistance, as shown in Fig. 7 (see also Eq. (9.2) in Sec. IX).

To make now a contact between the principal circuit, as depicted in Fig. 1 and governed by the Lagrangian (2.2), and the desired Jordan form J, we introduce its characteristic polynomial χs that has to be of the form

χs=detsI2nJ=s2n+1nk=1nanks2nk
(2.5)
=s2n+1nan1s2n1+an2s2n2+a0,

where parameters ak are real-valued and satisfy

<ak<,0kn1;a00.
(2.6)

The Jordan form J has to satisfy some a priori symmetry conditions to be associated with a Hamiltonian matrix H. In particular, its characteristic polynomial χs has to be even polynomial as indicated by Eq. (2.5), and its parameters ak must be as described in relations (2.6). The details on the indicated properties of matrices J and H are provided in Sec. VI.

We then relate the principal circuit to the characteristic polynomial χs by the setting up the following expressions for the circuit electric inductances, capacitances, and gyration resistances in terms of the coefficients ak of the polynomial χs:

Lj=11j1aj1,1jn,
(2.7)
Cj=1j1aj,Gj=11j1aj1jn1,Cn=1.
(2.8)

Note that Eqs. (2.7) and (2.8) imply the following identities:

CjLj+1=1,CjGj=1,1jn1,
(2.9)

as well as following expressions for coefficients aj in terms of the circuit parameters:

aj=11jLj+1,0jn1,aj=1j11Gj=1j1Cj,1jn1.
(2.10)

Under the assumptions that the circuit elements values satisfy Eqs. (2.7) and (2.8), the Lagrangian L defined by Eq. (2.2) is related to our principal Hamiltonian H defined by Eq. (5.3). The relationship between L and H is as follows: The Lagrangian L obtained from H by the Legendre transformation has exactly the same EL equations as the Lagrangian L (see Secs. VI and VIII A).

We show in  Appendix D that any solution to the EL equations (2.3) and (2.4) also satisfies the scalar differential equation

χtqkt=0,1kn,
(2.11)

indicating that the circuit Hamiltonian matrix H is cyclic and is determined by the characteristic polynomial χs defined by Eq. (2.5).

The principal circuit for two loops is shown in Fig. 2. It is the simplest case of our principal circuit that carries most of the significant properties of the general case. The general form (2.5) of the characteristic polynomial for n = 2 turns into

χs=s4+a1s2+a0.
(2.12)

The general form (2.2) of the principal circuit Lagrangian yields for n = 2

L=L1tq122+L2tq222q122C1q222C2+G1q1tq2q2tq12,
(2.13)

and the corresponding EL equations are

L1t2q1G1tq2+q1C1=0,L2t2q2+G1tq1+q2C2=0.
(2.14)

In particular, as the consequence of Eqs. (2.9)–(2.4) as well as the data in Table I, the following identities hold:

C1L2=1,C1G1=1;a0=1L1,a1=1L2=1G1=C1.
(2.15)
FIG. 2.

Principle circuit for 2 LC-loops.

FIG. 2.

Principle circuit for 2 LC-loops.

Close modal
TABLE I.

Circuit element values, n = 2.

k 
Lk 1a0 1a1 
Ck a1 −1 
Gk 1a1  
k 
Lk 1a0 1a1 
Ck a1 −1 
Gk 1a1  

The set of values of the principal circuit elements described by Eqs. (2.7) and (2.8) for n = 2 are listed in Table I. The significant circuit matrices in this case are as follows:

H=00a00100a100010100,C=010000100001a00a10,T=0a000a00a1000100001.
(2.16)

The determinants of the above matrices are as follows:

detH=detC=a0,detT=a02.
(2.17)

The similarity between matrices H and C takes here the form

T1HT=C
(2.18)

and can be verified by showing that HT=CT1 based on expressions (2.16) for the involved matrices.

Solutions to the EL equations (2.14) according to Eq. (2.6) satisfy the following scalar differential equation:

t4+a1t2+a0qkt=0,k=1,2.
(2.19)

We define special circuits as implementations of our principal circuit tailored to the specially chosen characteristic polynomials χs to achieve the desired Jordan forms. Namely, we are interested in

χs=s2a2n,s2+b2n,n=2,3,4,
(3.1)

for real a and b corresponding to the Jordan form J made of two Jordan blocks Jn±a and Jn±bi, respectively, where Jns is the n × n matrix defined by Eq. (1.1). In Sec. III E, we consider special circuits associated with the characteristic polynomials χs defined by Eq. (3.1) for arbitrary n ≥ 2.

We are also interested in

χs=sζsζ̄s+ζs+ζ̄2,ζ=a+bi,
(3.2)

where a≠0 and b ≠ 0 corresponding to the Jordan form J made of four Jordan blocks J2±a±bi and J2±abi.

When considering special circuits, we evaluate the significant matrices H, J, C, Y, Z, and T related by Eqs. (1.4) and (1.5) (see Sec. I F).

This is the simplest case demonstrating nontrivial Jordan forms, and for that reason, we study it greater detail. The special circuit shown in Fig. 2 has two f-loops. To get the desired Jordan form, we use one of the polynomials

χs=s2a22,s2+b22
(3.3)

and assign to the circuit elements the values provided in Table II [the parameter r refers to the roots of the characteristic polynomial χs, that is, χr=0].

TABLE II.

Circuit element values, n = 2.

k 
Lk 1a4 12a2 
Ck −2 a2 −1 
Gk 12a2  
r; n ±a; 2  
k 
Lk 1b4 12b2 
Ck 2 b2 −1 
Gk 12b2  
r; n ±bi; 2  
k 
Lk 1a4 12a2 
Ck −2 a2 −1 
Gk 12a2  
r; n ±a; 2  
k 
Lk 1b4 12b2 
Ck 2 b2 −1 
Gk 12b2  
r; n ±bi; 2  

In the case of the first polynomial χs that has real roots ±a, the circuit significant matrices are as follows:

H=00a401002a200010100,J=a1000a0000a1000a,T=0a400a402a2000100001,
(3.4)
C=010000100001a402a20,Y=1010a1a1a22aa22aa33a2a33a2,Z=a5a4a5a4a44a3a44a3a22aa22aa33a2a33a2.
(3.5)

In the case of the second polynomial χs that has pure imaginary roots ±bi, the significant circuit matrices are

H=00b401002b200010100,J=bi1000bi0000bi1000bi,T=0b400b402b2000100001,
(3.6)
C=010000100001b402b20,Y=1010bi1bi1b22bib22bib3i3b2b3i3b2,
(3.7)
Z=b5ib4b5ib4b44b3ib44b3ib22bib22bib3i3b2b3i3b2.
(3.8)

Applying to the case n = 2 the general formulas (3.34) for the eigenfrequencies ωj=±1LjCj of LC-loops as they were decoupled, we obtain

ω1=±b2,ω2=±2b.
(3.9)

This special circuit shown in Fig. 3 has three f-loops. It provides for two Jordan blocks of order 3 for circuit elements values as in Table III [the parameter r refers to the roots of the characteristic polynomial χs, that is, χr=0]. The corresponding polynomial is

χs=s2a23,s2+b23.
(3.10)

In the case when the roots of χs are real numbers ±a, the circuit matrices are

H=000a60010003a40010003a2000010000001001000,J=a100000a100000a000000a100000a100000a,
(3.11)
Z=a82a7a6a82a7a62a78a69a52a78a69a5a66a515a4a66a515a4a33a23aa33a23aa44a36a2a44a36a2a55a410a3a55a410a3.
(3.12)
FIG. 3.

Special circuit for three LC-loops.

FIG. 3.

Special circuit for three LC-loops.

Close modal
TABLE III.

Circuit element values, n = 3.

k 
Lk 1a6 13a4 13a2 
Ck −3 a4 −3 a2 −1 
Gk 13a4 13a2  
r; n ±a; 3   
k 
Lk 1b6 13b4 13b2 
Ck −3 b4 3 b2 −1 
Gk 13b4 13b2  
r; n ±bi; 3   
k 
Lk 1a6 13a4 13a2 
Ck −3 a4 −3 a2 −1 
Gk 13a4 13a2  
r; n ±a; 3   
k 
Lk 1b6 13b4 13b2 
Ck −3 b4 3 b2 −1 
Gk 13b4 13b2  
r; n ±bi; 3   

In the case when the roots of χs are pure imaginary numbers ±bi, the circuit matrices are

H=000b60010003b40010003b2000010000001001000,J=bi100000bi100000bi000000bi100000bi100000bi,
(3.13)
Z=b82b7ib6b82b7ib62b7i8b69b5i2b7i8b69b5ib66b5i15b4b66b5i15b4b3i3b23bib3i3b23bib44b3i6b2b44b3i6b2b5i5b410b3ib5i5b410b3i.
(3.14)

Applying to the case n = 3 the general formulas (3.34) for the eigenfrequencies ωj=±1LjCj of LC-loops as they were decoupled, we obtain

ω1=±b3,ω2=±b,ω3=±3b.
(3.15)

This special circuit shown in Fig. has four f-loops. It provides for two Jordan blocks of order 4 for circuit elements values as in Tables IV and V.

TABLE IV.

Circuit element values, n = 4.

k 
Lk 1a8 14a6 16a4 14a2 
Ck −4 a6 −6 a4 −4 a2 −1 
Gk 14a6 16a4 14a2  
r; n ±a; 4    
k 
Lk 1a8 14a6 16a4 14a2 
Ck −4 a6 −6 a4 −4 a2 −1 
Gk 14a6 16a4 14a2  
r; n ±a; 4    
TABLE V.

Circuit element values, n = 4.

k 
Lk 1b8 14b6 16b4 14b2 
Ck 4 b6 −6 b4 4 b2 −1 
Gk 14b6 16b4 14b2  
r; n ±bi; 4    
k 
Lk 1b8 14b6 16b4 14b2 
Ck 4 b6 −6 b4 4 b2 −1 
Gk 14b6 16b4 14b2  
r; n ±bi; 4    

The values of circuit elements are provided in Tables IV and V [the parameter r refers to the roots of the characteristic polynomial χs, that is, χr=0].

The corresponding polynomial is

χs=s2a24,s2+b24.
(3.16)

In the case when the roots of χs are real numbers ±a, the circuit matrices are

H=0000a8000100004a6000100006a4000100004a200000100000000100000000100010000,J=a10000000a10000000a10000000a00000000a10000000a10000000a10000000a,
(3.17)
Z=a113a103a9a8a113a103a9a83a1014a923a816a73a1014a923a816a73a919a848a756a63a919a848a756a6a88a728a656a5a88a728a656a5a44a36a24aa44a36a24aa55a410a310a2a55a410a310a2a66a515a420a3a66a515a420a3a77a621a535a4a77a621a535a4.
(3.18)

In the case when the roots of χs are pure imaginary numbers ±bi, the circuit matrices are

H=0000b8000100004b6000100006b4000100004b200000100000000100000000100010000,J=bi10000000bi10000000bi10000000bi00000000bi10000000bi10000000bi10000000bi,
(3.19)
Z=b11i3b103b9ib8b11i3b103b9ib83b1014b9i23b816b7i3b1014b9i23b816b7i3b9i19b848b7i56b63b9i19b848b7i56b6b88b7i28b656b5ib88b7i28b656b5ib44b3i6b24bib44b3i6b24bib5i5b410b3i10b2b5i5b410b3i10b2b66b5i15b420b3ib66b5i15b420b3ib7i7b621b5i35b4b7i7b621b5i35b4.
(3.20)

Applying to the case n = 4 the general formulas (3.34) for the eigenfrequencies ωj=±1LjCj of LC-loops as they were decoupled, we obtain

ω1=±b2,ω2=±63b,ω3=±62b,ω4=±2b.
(3.21)

The special circuit for four f-loops is shown in Fig. 4. The circuit polynomial in this special case is

χs=sζsζ̄s+ζs+ζ̄2=a2+b2+2bs+s22a2+b22bs+s22,ζ=a+bi,
(3.22)

where a ≠ 0 and b ≠ 0. The values of the circuit elements are provided in Table VI [the parameter r refers to the roots of the characteristic polynomial χs, that is, χr=0]. We have computed all significant matrices, but the number of their entries combined with the length of their expressions are too large to be displayed here.

FIG. 4.

Special circuit for three LC-loops.

FIG. 4.

Special circuit for three LC-loops.

Close modal
TABLE VI.

Circuit element values, n = 2.

k 
Lk 1a2+b24 14a2b2a2+b22 123a4+3b4a2b2 14a2b2 
Ck 4a2b2a2+b22 23a4+3b4a2b2 4a2b2 −1 
Gk 14a2b2a2+b22 122a2b23a43b4 14a2b2  
r; n a ± bi, −a ± bi; 2    
k 
Lk 1a2+b24 14a2b2a2+b22 123a4+3b4a2b2 14a2b2 
Ck 4a2b2a2+b22 23a4+3b4a2b2 4a2b2 −1 
Gk 14a2b2a2+b22 122a2b23a43b4 14a2b2  
r; n a ± bi, −a ± bi; 2    

We consider here the special circuits associated with the characteristic polynomials χs defined by Eq. (3.1) for arbitrary n ≥ 2. Namely, in the case of χs=s2a2n using the binomial formula and expression (2.5) for the general characteristic polynomial, we obtain the following expressions for coefficients:

aj=1jnja2nj,1jn,
(3.23)

where nj is the binomial coefficient defined in Eq. (F7). From, Eqs. (2.7) and (2.8), we obtain the following formulas for the circuit elements:

Lj=11j1aj1=1nj1a2nj+1,1jn,
(3.24)
Cj=1j1aj=nja2nj,1jn1,Cn=1,
(3.25)
Gj=11j1aj=1nja2nj,1jn1.
(3.26)

Equations (3.24) and (3.25) imply

LjCj=njnj1a2=nj+1ja2,1jn.
(3.27)

Formulas (3.27) imply, in turn, the following expressions for the eigenfrequencies of involved LC-loops when decoupled:

ωj=±1LjCj=±jnj+1ai,1jn.
(3.28)

The eigenfrequencies ωj in Eq. (3.28) are evidently pure imaginary.

The values of circuit elements and other quantities associated with the characteristic polynomial χs=s2+b2n can be readily obtained from Eqs. (3.23)–(3.28) by plugging into them a = bi. This yields

aj=1nnjb2nj,1jn,
(3.29)
Lj=11j1aj1=1nj+1nj1b2nj+1,1jn,
(3.30)
Cj=1j1aj=1nj+1njb2nj,1jn1,Cn=1,
(3.31)
Gj=11j1aj=1nj+1njb2nj,1jn1,
(3.32)
LjCj=njnj1b2=nj+1jb2,1jn.
(3.33)

Then, the eigenfrequencies of LC-loops as they were decoupled are

ωj=±1LjCj=±jnj+1b,1jn.
(3.34)

In Secs. II and III, our primary goal was to introduce and study some circuits with evolution matrices exhibiting nontrivial Jordan blocks. The goal of this section is somewhat different. We study here our simplest circuit composed of two LC-loops coupled by a gyrator, as shown in Fig. 2, without imposing initially any a priori assumptions on the circuit parameters L1, C1, L2, C2, and G1 except for that they are all real and non-zero. In particular, we derive in this section the most general conditions on these parameters under which the relevant evolution matrix exhibits nontrivial Jordan blocks. The Lagrangian L for such a circuit is described by Eq. (2.13), and its evolution equations are the corresponding EL equations (2.14). These equations can be readily recast into the following matrix form:

Atq=0,q=q1q2,As=s2+1L1C1sG1L1sG1L2s2+1L2C2,
(4.1)

where As is evidently a 2 × 2 monic matrix polynomial of s of degree 2, namely,

As=s21001+s0G1L1G1L20+1L1C1001L2C2.
(4.2)

Then, the matrix polynomial eigenvalue problem associated with the matrix differential equation (4.1) is

Asq=s2+1L1C1sG1L1sG1L2s2+1L2C2q1q2=0.
(4.3)

The matrix polynomial eigenvalue problem (4.3) is evidently nonlinear. According to matrix polynomial theory reviewed in  Appendix C, the second-order vector differential equation (4.1) can be reduced to the standard first-order vector differential equation

tq=Cq,C=001000011L1C100G1L101L2C2G1L20,q=qtq,
(4.4)

where C is the 4 × 4 companion matrix for the matrix polynomial As. Then, the standard eigenvalue problem corresponding to the matrix polynomial eigenvalue problem (4.2) is

sICq=0,q=qsq.
(4.5)

The characteristic polynomial associated with the matrix polynomial As and its linearized version sIC is

χs=detAs=detsIC=s4+G12L1L2+1L2C2+1L1C1s2+1L1C1L2C2.
(4.6)

Consequently, the eigenvalues associated with the eigenvalue problems (4.3) and (4.5) can be found from the characteristic equation

χs=s4+G12L1L2+1L2C2+1L1C1s2+1L1C1L2C2=0.
(4.7)

Our primary goal here is to identify all real non-zero values of the circuit parameters L1, C1, L2, C2, and G1 for which the matrix C defined by Eq. (4.4) has the nontrivial Jordan form. Our general studies of matrix polynomials in  Appendix C, particularly Theorem 26, imply the following statement:

Theorem 1
(Jordan form of the companion matrix). Lets0be an eigenvalue of the companion matrixCdefined byEq. (4.4)such that its algebraic multiplicityms02. Then (i)s0 ≠ 0; (ii)s0is also an eigenvalue ofC; (iii)s0is either real or pure imaginary; (iv)ms0=ms0=2; and (v) the Jordan formJof the matrixCis
J=s01000s00000s01000s0.
(4.8)
That is, because of the special form the of companion matrixC, the eigenvalue degeneracy forCimplies that its Jordan formJconsists of two Jordan blocks of size 2.

Proof.

The eigenvalue s0 satisfies χs0=0. Since in view of Eq. (4.6), χ0=1L1C1L2C20, we infer that s0 ≠ 0. The characteristic equation (4.7) implies that χs0=χs0=0, and hence, −s0 is an eigenvalue. Note that since all coefficients of the characteristic equation (4.7) are real, then the number s̄0, which is complex-conjugate to s0, is also an eigenvalue since χs̄0=χs0̄=0. If s0 would be a complex number with non-zero real and imaginary parts, then there would be four distinct eigenvalues s0, −s0, s̄0, and s̄0 for the fourth-degree characteristic equation. This would make it impossible for the algebraic multiplicity of s0 to satisfy ms02, which is a condition of the theorem. Hence, we have to infer that s0 is either real or pure imaginary. Consequently, s0 and −s0 are the only eigenvalues and are the roots of the characteristic polynomial χs. Since χs involves only even degrees of s, we also have ms0=ms0, implying ms0=ms0=2.

Note now that
As0 for any complex s,
(4.9)
and otherwise, based on entries of matrix As defined by Eq. (4.1), we will have to infer consequently that s = 0 and then 1L1C1=0, but the later is impossible. Note also
dimkerAs0=1.
(4.10)
Indeed, since detAs0=0, we have dimkerAs01. On the other hand, in view of relation (4.9), As00, implying that dimkerAs0<2. Hence, we may conclude that Eq. (4.10) holds. Using Eq. (4.10) and the statement of Theorem 26, we can infer that the matrix C has exactly one Jordan block associated with s0 of the size ms0=2, and the same statement holds for −s0. Consequently, the Jordan form of matrix C satisfies Eq. (4.8), and this completes the proof of the theorem.□

The further analytical developments suggest introducing the following variables:

ξ1=1L1C1,ξ2=1L2C2,g=G12,h=s2
(4.11)

and refer to positive g as the gyration parameter. Then, the companion matrix C defined by Eq. (4.4) and its characteristic function χs as in Eq. (4.7) take, respectively, the following forms:

C=00100001ξ100G1L10ξ2G1L20,
(4.12)
χs=χh=h2+ξ1+ξ2+gL1L2h+ξ1ξ2,h=s2.
(4.13)

Being interested in degenerate eigenvalues satisfying the equation χs=χh=0, we turn to the discriminant Δh of the quadratic polynomial χh defined by Eq. (4.13), namely,

Δh=g2L12L22+2ξ1+ξ2gL1L2+ξ1ξ22.
(4.14)

Recall that the solutions to the quadratic equation χh = 0 are

h±=ξ1+ξ2+gL1L2±Δh2.
(4.15)

Then, the corresponding four solutions s to the characteristic equation (4.7), that is, the eigenvalues, are

s=±h+,±h,
(4.16)

where h± satisfy Eq. (4.15).

Turning back to h± in (4.15), we note that the eigenvalue degeneracy condition turns into the equation Δh = 0. This equation can be viewed a constraint on the coefficients of the quadratic in the h polynomial χh and ultimately on the circuit parameters, namely,

L12L22Δh=g2+2ξ1+ξ2gL1L2+ξ1ξ22L12L22=0.
(4.17)

Equation (4.17) is evidently a quadratic equation for g. Being given remaining circuit coefficients ξ1, ξ2, L1, and L2, this quadratic in the g equation has exactly two solutions,

gδ=ξ1ξ2+2δξ1ξ2L1L2,δ=±1.
(4.18)

We refer to gδ in Eq. (4.18) as special values of the gyration parameterg. For the two special values ġ, we get from Eq. (4.15) the corresponding two degenerate roots

h=ξ1+ξ2+ġL1L22=±ξ1ξ2.
(4.19)

Since G1 is real, then g=G12 is real as well. Expression (4.18) for g is real-valued if and only if

ξ1ξ2>0, or equivalently ξ1ξ1=ξ2ξ2=σ,
(4.20)

where we introduced a binary variable σ taking values ±1. We refer to σ as the circuit sign index. Relations (4.20) imply, in particular, that the equality of signs signξ1=signξ2 is a necessary condition for the eigenvalue degeneracy condition Δh = 0, provided that g has to be real-valued.

It follows then from relations (4.18) and (4.20) that the special values of the gyration parameter gδ can be recast as

gδ=σξ1+δξ22L1L2,δ=±1,
(4.21)

where ξ>0 for ξ > 0. Recall that g=G12>0, and to provide for that, we must have in the right-hand side of Eq. (4.21)

σL1L2>0, or equivalently L1L2L1L2=σ.
(4.22)

Relations (4.20) and (4.22) on the signs of the involved parameters can be combined into the circuit sign constraints

signξ1=signξ2=signL1L2=signσ.
(4.23)

Note that the sign constraints(4.23)involving the circuit indexσdefined by(4.20)are necessary for the eigenvalue degeneracy condition Δh = 0. Combining Eqs. (4.21) and (4.22), we obtain

gδ=ξ1+δξ22L1L2,δ=±1, assuming the circuit sign constraints.
(4.24)

Since g=G12>0, the special values of the gyrator resistance Ġ1 corresponding to the special values gδ as in Eq. (4.24) are

Ġ1=σ1gδ=σ1ξ1+δξ2L1L2, assuming the circuit sign constraints, 
(4.25)

where the binary variable σ1 takes values ±1.

Using representation (4.24) for gδ under the circuit sign constraints (4.23), we can recast the expression for the degenerate root ḣ in Eq. (4.19) as follows:

hδ=σδξ1ξ2,  for g=gδ=ξ1+δξ22L1L2,δ=±1,
(4.26)

where σ is the circuit sign index defined by Eq. (4.20) and ξ>0 for ξ > 0.

The important elements of the above analysis are summarized in the following statement:

Lemma 2

(the signs of the circuit parameters). Let the circuit be as depicted inFig. 2and all its parametersL1,C1,L2,C2, andG1be real and non-zero. Then, the companion matrixCsatisfying equations (4.4) and (4.12)has a degenerate eigenvalue if and only if (i) the sign constraints(4.23)hold and (ii)g=G12satisfiesEq. (4.24). Under the circuit sign constraints(4.23), the degenerate solutionhδto the quadratic equationχh = 0 [seeEq. (4.13)] is determined byEq. (4.26).

The sign constraints (4.23) are equivalent to
signC1=signL2,signL1=signC2.
(4.27)

Proof.

The statements of lemma before Eq. (4.27) have been already argued. The verification of the equivalency of Eqs. (4.23) and (4.27) is straightforward.□

Note that when Lemma 2 provides sharp criteria for the companion matrix C to have a degenerate eigenvalue, Theorem 1 states that if the companion matrix C has such an eigenvalue, then its Jordan form J is formed by two Jordan blocks as in Eq. (4.8). Consequently, the following statement holds that combines statements of Theorem 1 and Lemma 2.

Theorem 3
(Jordan form under degeneracy). Let the circuit be as depicted inFig. 2, and let all its parametersL1,C1,L2,C2, andG1be real and non-zero. Then, the companion matrixCsatisfying (4.4) and (4.12) has the Jordan form
J=s01000s00000s01000s0
(4.28)
if and only if the circuit parameters satisfy the degeneracy conditions described in Lemma 2. Then, forg = gδ, we have
hδ=σδξ1ξ2, for g=gδ=ξ1+δξ22L1L2,
(4.29)
±s0=±σδξ1ξ2, for g=gδ=ξ1+δξ22L1L2,
(4.30)
whereξ>0forξ > 0,δ = ±1, andσis the circuit sign index defined byEq. (4.20). According to formula(4.30), degenerate eigenvalues ±s0depend on the productσδandξ1,ξ2, and consequently, they are either real or pure imaginary depending on whetherδ = σorδ = −σ.
Note that in the special case whenξ1=ξ2, the parametergδtakes only one non-zero value, namely,
g=g1=4ξ1L1L2,L1L2=σL1L2,
(4.31)
whereasg−1 = 0, which is inconsistent with our assumptionG1 ≠ 0. Evidently, forg = 0, the circuit breaks into two independentLC-circuits, and in this case, the relevant Jordan form is a diagonal 4 × 4 matrix with eigenvalues±ξ1and±ξ2.

Remark 4

(instability and marginal stability). Note that according to formula (4.30), degenerate eigenvalues ±s0 are real for δ = σ, and hence, they correspond to exponentially growing and decaying in time solutions indicating instability. For δ = −σ, the degenerate eigenvalues ±s0 are pure imaginary corresponding to oscillatory solutions, indicating that there is at least marginal stability.

To get a graphical illustration for the circuit complex-valued eigenvalues as functions of the gyration parameter g, we use the following data:

ξ1=1,ξ2=2,L1=1,L2=2,σ=1,
(4.32)

and this corresponds to

ξ1=1,ξ2=2,L1=±1,L2=2.
(4.33)

It follows then from representation (4.24) that the corresponding special values gδ are

g1=21220.3431457498,g1=21+2211.65685425.
(4.34)

To explain the rise of the circular part of the set Seig in Fig. 5, we recast the characteristic equation (4.13) as follows:

H+1H=R,R=σξ1ξ2gL1L2ξ1+ξ2,H=hξ1ξ2.
(4.35)

Note that

R=2δσ,  for  g=gδ=ξ1+δξ22L1L2,δ=±1,
(4.36)

where σ = ±1 is the circuit sign index. Since R depends linearly on g, relations (4.35) and (4.36) imply

R2, for g1gg1;R>2, for g<g1 and g>g1.
(4.37)

It is an elementary fact that solutions H to Eq. (4.35) satisfy the following relations:

H=expiθ, for R2, and cosθ=R2,0θπ,
(4.38)
H>0, for R>2, and H<0, for R<2.
(4.39)

It is also evident from the form of Eq. (4.35) that if H is its solution, then H−1 is a solution as well, that is, the two solutions to Eq. (4.35) always come in pairs of the form H,H1.

FIG. 5.

The plot shows the set Seig of all complex-valued eigenvalues s defined by Eq. (4.16) for the data in Eqs. (4.32) and (4.33) when the gyration parameter g varies in the interval containing special values g−1 and g1 defined in relations (4.34). The horizontal and vertical axes represent the real and the imaginary parts Rs and Is of eigenvalues s. Seig consists of the circle centered in the origin of radius ξ1ξ24 and four intersecting intervals lying on real and imaginary axes. The circular part of the set Seig corresponds to all eigenvalues for g−1gg1. The degenerate eigenvalues ±s0 corresponding to g−1 and g1 and defined by Eq. (4.30) are shown as solid diamond (blue) dots. Two of them are real, positive and negative, numbers, and the other two are pure imaginary, with positive and negative imaginary parts. 16 solid circular (red) dots are associated with four quadruples of eigenvalues corresponding to four different values of the gyration parameter g chosen to be slightly larger or smaller than the special values g−1 and g1. Let us take a look at any of the degenerate eigenvalues identified by solid diamond (blue) dots. If g is slightly different from its special values g−1 and g1, then each degenerate eigenvalue point splits into a pair of points identified by solid circular (red) dots. They are either two real or two pure imaginary points if g is outside the interval g1,g1, or alternatively, they are two points lying on the circle if g is inside the interval g1,g1.

FIG. 5.

The plot shows the set Seig of all complex-valued eigenvalues s defined by Eq. (4.16) for the data in Eqs. (4.32) and (4.33) when the gyration parameter g varies in the interval containing special values g−1 and g1 defined in relations (4.34). The horizontal and vertical axes represent the real and the imaginary parts Rs and Is of eigenvalues s. Seig consists of the circle centered in the origin of radius ξ1ξ24 and four intersecting intervals lying on real and imaginary axes. The circular part of the set Seig corresponds to all eigenvalues for g−1gg1. The degenerate eigenvalues ±s0 corresponding to g−1 and g1 and defined by Eq. (4.30) are shown as solid diamond (blue) dots. Two of them are real, positive and negative, numbers, and the other two are pure imaginary, with positive and negative imaginary parts. 16 solid circular (red) dots are associated with four quadruples of eigenvalues corresponding to four different values of the gyration parameter g chosen to be slightly larger or smaller than the special values g−1 and g1. Let us take a look at any of the degenerate eigenvalues identified by solid diamond (blue) dots. If g is slightly different from its special values g−1 and g1, then each degenerate eigenvalue point splits into a pair of points identified by solid circular (red) dots. They are either two real or two pure imaginary points if g is outside the interval g1,g1, or alternatively, they are two points lying on the circle if g is inside the interval g1,g1.

Close modal

Since the eigenvalues s satisfy s=±h, the established above properties of h=ξ1ξ2H can recast for s as follows:

Theorem 5
(quadruples of eigenvalues). For everyg > 0, every solutionsto the characteristic equation (4.7)is of the form(4.16)and the number of solutions is exactly four, counting their multiplicity. Every such a quadruple of solutions is of the following form:
s,ξ1ξ2s,s,ξ1ξ2s,
(4.40)
wheresis a solution to the characteristic equation (4.7). Then, forg−1gg1, the quadruple of solutions belongs to the circles=ξ1ξ24such that
s=δ1ξ1ξ24expiδ2θ,cosθ=R2,0θπ,δ1,δ2=±1,
(4.41)
whereRis defined in relations(4.35). Ifg < g−1org > g1, the quadruple of solutions consists of either real numbers or pure imaginary numbers ifR > 2 orR < −2, respectively. In view of relations(4.35)ands=±h, whereh=ξ1ξ2H, every quadruple of solutions as in expression(4.40)is invariant with respect to the complex conjugation transformation.

The following remark discusses in some detail the transition of eigenvalues lying on the circle s=ξ1ξ24 having non-zero real and imaginary parts into either real or pure imaginary numbers as the value of the gyration parameter g passes through its special values g−1 or g1 at which the eigenvalues degenerate.

Remark 6

(transition at degeneracy points). According to formula (4.30), there is a total of four degenerate eigenvalues ±s0, namely, ±ξ1ξ24 and ±iξ1ξ24 [depicted as solid diamond (blue) dots in Fig. 5], that are associated with the two special values of the gyration parameter g±1=ξ1±ξ22L1L2. For any value of the gyration parameter g different than its two special values, there are exactly four distinct eigenvalues s forming a quadruple as in expression (4.40). If g−1 < g < g1 and g gets close to either g−1 or g1, the corresponding four distinct eigenvalues on the circle s=ξ1ξ24 get close to either ±ξ1ξ24 or ±iξ1ξ24 (as depicted in Fig. 5) by solid circle (red) dots. As g approaches the special values g−1 or g1, reaches them, and gets out of the interval g1,g1, the corresponding solid circle (red) dots approach the relevant points ±ξ1ξ24 or ±iξ1ξ24, merge at them, and then split again passing to, respectively, real and imaginary axes, as illustrated by Fig. 5.

Theorem 1 provides a general statement that the degeneracy of the companion matrix C defined by Eq. (4.4) implies that its Jordan form consists of two Jordan blocks as in Eq. (4.8). We would extend that statement with a construction of the corresponding Jordan basis. With that in mind, we introduce the following matrix evidently related to the companion matrix C:

C=Cξ1,ξ2,b1,b2=00100001ξ100b10ξ2b20,
(4.42)

Namely,

C=Cξ1,ξ2,b1,b2,b1=G1L1,b2=G1L2.
(4.43)

Note that the change of the sign of G1 of the gyration capacitance or change of the sign of the parameters b1, b2 in matrix C yields a matrix that is similar to the original matrix, that is, if C is a matrix defined by Eq. (4.42), we have

Cξ1,ξ2,b1,b2=Λ1Cξ1,ξ2,b1,b2Λ,Λ=1000010000100001.
(4.44)

Equations (4.43) and (4.44) readily imply the following statement:

Lemma 7
(alteration of the gyration resistance). Let the circuit be as described in Theorem 3. Then, the sign alteration of the gyration resistanceG1yields a circuit with the evolution matrixCG1that is similar to the evolution matrixCG1of the original circuit, that is,
CG1=Λ1CG1Λ,Λ=1000010000100001.
(4.45)

The above considerations suggest introducing some two special form matrices intimately related to the companion matrix C defined by expression (4.12). It is a tedious but straightforward exercise to verify that the following statements hold for these matrices:

Lemma 8
(Jordan form of a special matrix). LetC±be 4 × 4 matrices of the form
C±=C±ζ1,ζ2,b1,b2=00100001ζ1200b10ζ22b20, where  b1b2=ζ1±ζ220,
(4.46)
whereζ1, ζ2andb1, b2are complex numbers. Then, matricesC±can be recast as
C±=C±ζ1,ζ2,α=00100001ζ1200αζ1±ζ20ζ22ζ1±ζ2α0,α=b1ζ1±ζ2.
(4.47)
The Jordan formsJ±of the corresponding matricesC±are
J±=J±ζ1,ζ2=Z±1C±Z±=±ζ1ζ21000±ζ1ζ20000±ζ1ζ21000±ζ1ζ2,
(4.48)
where±ζ1ζ2is one of the values of the square root of ±ζ1ζ2, and matricesZ±are
Z±=Z±ζ1,ζ2,α=ζ14±ζ1ζ212ζ1±ζ2ζ14±ζ1ζ212ζ1±ζ2ζ14ζ2α±ζ14ζ2α±ζ1ζ2ζ14ζ2αζ14ζ2α±ζ1ζ2ζ14ζ1ζ1ζ24ζ1±ζ2±ζ1ζ2ζ14ζ1ζ1ζ24ζ1±ζ2±ζ1ζ2ζ124α±ζ1ζ20ζ124α±ζ1ζ20.
(4.49)
Note that the columns of matricesZ±form the Jordan bases of the corresponding matricesC±, and the first and the third columnsZ±are the eigenvectors of the corresponding matricesC±with respective eigenvalues±ζ1ζ2and±ζ1ζ2.
Note also that the matricesC±andZ±can be factorized as follows:
C±ζ1,ζ2,α=Λα1C±ζ1,ζ2,1Λα,Z±ζ1,ζ2,α=Λα1Z±ζ1,ζ2,1,
(4.50)
C±ζ1,ζ2,b1,b2=Λ11C±ζ1,ζ2,b1,b2Λ1,
(4.51)
where matrix Λαis the following diagonal matrix:
Λα=10000α000010000α.
(4.52)

In the case when the gyration resistance takes its special values Ġ1 as in Eq. (4.25), the companion matrix C defined by expression (4.12) can be cast as matrix C± in Eq. (4.46). Indeed, using Eq. (4.25), we readily obtain

Ġ12L1L2=σξ1+δξ22=ζ1+δζ22, where ξ>0,
(4.53)
ζj=ξjforσ=1iξjforσ=1,j=1,2.

Combining Eq. (4.53) with Lemma 8, we arrive at the following statement:

Theorem 9
(degeneracy and the Jordan form). Let the circuit be as depicted inFig. 2, and let all its parametersL1,C1,L2,C2, andG1be real and non-zero. Then, the companion matrixCdefined by expression(4.12)has a degenerate eigenvalue if and only if (i) the sign constraints(4.23)are satisfied and (ii) its gyration parametergtakes its two special values,
gδ=ξ1+δξ22L1L2,δ=±1,
(4.54)
and the corresponding special values of the gyration resistance,
Ġ1=σ1gδ=σ1ξ1+δξ2L1L2,
(4.55)
where the binary variableσ1takes values ±1. For these special values of the gyration resistance, matrixChas exactly two degenerate eigenvalues ±s0of multiplicity 2, satisfying the following equations:
±s0=±σδξ1ξ2,
(4.56)
whereξ>0forξ > 0,δ = ±1, andσ = ±1 is the circuit sign index defined byEq. (4.20). In addition to that, matrixCcan be represented as matrixC±inEq. (4.47), withζ1, ζ2described byEq. (4.53)and
α=L1L1σ1L2L1forσ=1L1L1σ1iL2L1forσ=1.
Consequently, all statements of Lemma 8 for matrixC±hold including its Jordan formJ±as inEq. (4.48)and expressions(4.49)for the Jordan basis as columns of matrixZ±.

The first goal of our synthesis process is to construct a Hamiltonian system governed by the evolution equation (1.2) with the circuit matrix H having the prescribed Jordan canonical form subject to natural constraints. Consequently, the circuit matrix H has to be a Hamiltonian matrix, that is, a matrix obtained from a quadratic Hamiltonian H with real coefficients. The spectrum specH, that is, the set of all distinct eigenvalues, of a Hamiltonian matrix must have the following property:

if ζspecH, then ζ,ζ̄,ζ̄specH,the multiplicity of all four numbers is the same,
(5.1)

where ζ̄ stands for complex-conjugate to the complex number ζ (see Sec. VIII F for details). We refer to property (5.1) as the Hamiltonian spectral symmetry. Suppose that a ≠ 0 and b ≠ 0 are real numbers. Note then that the set ζ,ζ,ζ̄,ζ̄ consists of (i) two numbers a,a if ζ = a, (ii) two numbers bi,bi if ζ = bi, and (ii) four numbers

a+bi,abi,abi,a+bi
(5.2)

if ζ = a + bi.

To achieve the desired Jordan form for the circuit matrix H, we introduce the characteristic polynomial χHs=χJs and find its coefficients. Having coefficients ak of the polynomial χHs as in Eqs. (2.5) and (2.6), we define the Hamiltonian Ha by the following explicit expression:

Ha=k=1npk+1qk+12k=1n1k1ak1pk2+12qn2,
(5.3)
<ak<,0kn1,a00.
(5.4)

Note that the system parameters ak can be negative and positive. The particular choice of signs in expression (5.3) is a matter of convenience. The HamiltonianHadefined byEq. (5.3)is fundamental to the synthesis of all special circuits we construct, and we refer to it as the principal Hamiltonian.

The principal Hamiltonian matrix Ha that corresponds to the principal Hamiltonian Ha has the following properties (see Sec. VI for details):

  • The corresponding to Ha Hamiltonian matrix Ha has the polynomial χs defined by Eq. (2.5) as its characteristic polynomial χHas, and consequently, the set of the distinct roots sj of the polynomial is exactly the set of all distinct eigenvalues of the circuit matrix Ha, that is, specHa=sj.

  • Since a0 ≠ 0, we have sj ≠ 0 for every j.

  • The spectrum specHa satisfies Hamiltonian spectral symmetry condition (5.1).

  • The circuit matrix Ha is cyclic (nonderogatory), that is, the geometric multiplicity of every eigenvalue sj is exactly one, and every sj is associated with the single Jordan block Jnjsj of the size nj, which is the algebraic multiplicity of eigenvalue sj; in other words, there is always a single Jordan block for each distinct eigenvalue; the cyclicity property is an integral part of the construction yielding simpler Jordan forms.

  • If a non-zero ζspecHa is real or pure imaginary, then the Jordan form Ja of the Hamiltonian matrix Ha has two Jordan blocks Jnζ and Jnζ of the matching size n, where n is the multiplicity of ζ as the root of the polynomial χs.

  • If ζspecHa and ζ = a + bi, with a ≠ 0 and b ≠ 0, then the Jordan form Ja of the Hamiltonian matrix Ha has four Jordan blocks Jn±a±bi and Jn±abi of the matching size n, where n is the multiplicity of ζ as the root of the polynomial χs.

Making particular choices of ak for the principal Hamiltonian Ha allows us to achieve the desired Jordan forms. With that in mind, we introduce the following specific polynomials for real numbers non-zero numbers a and b:

χs=s2a2n,s2+b2n,
(5.5)
χs=s4+2b2a2s2+a2+b22n.
(5.6)

Note that polynomials in Eq. (5.5) have, respectively, two real roots ±a and two pure imaginary roots ±bi of multiplicities n, whereas the polynomial in Eq. (5.6) has four roots ±a ± bi and ±abi of multiplicities n. The Jordan forms Ja of system matrices Ha associated with the polynomials in Eqs. (5.5) and (5.6) are, respectively,

Ja=Jnζ00Jnζ,Ja=Jnζ0000Jnζ0000Jnζ̄0000Jnζ̄,ζ=a,bi,
(5.7)

where the Jordan block Jnζ is defined by Eq. (1.1).

We summarize now the important points of the analysis in Secs. II and III in the following statement:

Theorem 10

(principal circuit). Suppose that the principal circuit depicted inFig. 1has its element values defined by equations (2.7) and (2.8). Then, the dynamics of the principal circuit is governed by the principal LagrangianLdefined byEq. (2.2)and the principal HamiltonianHdefined by defined byEq. (5.3). The corresponding EL equations (2.3) and (2.4) represent the Kirchhoff voltage law, whereas the Kirchhoff current is enforced by the selection ofninvolved f-loops and currentstqk.

The relevant Hamiltonian matrixHis cyclic (non-derogatory), and its characteristic polynomialχsis defined byEq. (2.5). The Jordan formJof matrixHis completely determined byχs. In particular, each distinct rootsjofχsof the multiplicitynjis represented inJby the single Jordan blockJnjsjof the matching sizenj.

For particular choices of the monic polynomialχsas described in Eqs. (5.5) and (5.6), one obtains circuits associated with the Jordan forms represented, respectively, in Eq. (5.7).

Suppose that the system configuration is described by time-dependent n-dimensional vector-column q and its dynamics is governed by a Hamiltonian H=Hp,q, where p is the system momentum, which is an n-dimensional vector-column just as the configuration vector q. Suppose now that the Hamiltonian H is defined by Eq. (5.3). To present the system information in a compact matrix form, we recast the representation of Hamiltonian (5.3) as

Ha=12XTMHX,X=qp,MH=πnKnKnTDa,
(6.1)

where Da and πn are diagonal n × n matrices defined by

Da=a00000a100001n3an200001n2an1,πn=0000000000000001,
(6.2)

and Kn is the n × n nilpotent matrix defined by

Kn=0100001000010000.
(6.3)

We also make use of the Jordan block Jnζ of the size n defined by

Jnζ=ζIn+Kn=ζ1000ζ1000ζ1000ζ,
(6.4)

where In is the n × n identity matrix.

The evolution equations for the principal Hamiltonian Ha, defined by Eq. (6.1), are

tX=HaX,Ha=JMH,J=0II0,
(6.5)

where the system state vector X and matrix MH are defined by (5.3), and consequently,

Ha=JMH=KnTDaπnKn.
(6.6)

With an eigenvalue problem in mind, we introduce the matrix

sI2nHa=JnTsDaπnJns,
(6.7)

and then find that the corresponding characteristic function is equal to

χas=detsI2nHa=s2n+1nk=1nanks2nk=s2n+1nan1s2n1+an2s2n2+a0.
(6.8)

To see that representation (6.8) for χas holds, we apply formula (E3) to the right-hand side of Eq. (6.7) and obtain

χas=detsI2nHa=detJnTsdetJns+πnJn1sTDa.
(6.9)

We then use Eqs. (6.2) and (6.18) to evaluate the right-hand side of Eq. (6.9) and arrive at formula (6.8).

We introduce now the so-called companion to the polynomial χas (see  Appendix B), which is the 2n × 2n matrix defined by

Ca=010000010000000000000100000001c00c10cn10,ck=1n1ak,0kn.
(6.10)

Note that the eigenvalue problem for the companion matrix Ca has the following explicit form solution (see  Appendix B):

CaYs=sYs,Ys=1ss2s2n2s2n1,Yk=sk1,1k2n,χas=0,
(6.11)

where evidently the vector polynomial Ys is uniquely determined by the corresponding eigenvalue s. If all eigenvalues sj, 1 ≤ j ≤ 2n, of the companion matrix Ca are different, the set of the corresponding eigenvectors Ysj form a basis that diagonalize matrix Ca. In the general case, we introduce a 2n × 2n matrix Ya as the generalized Vandermonde matrix defined by equations (B10) and (B11). Then, according to Proposition 22, we have

Ca=YaJaYa1,
(6.12)

where Ja is the Jordan form of the companion matrix Ca. We refer to Ya as the Jordan basis matrix for matrix Ca. In the special case of distinct eigenvalues, matrix Ya turns into the standard Vandermonde matrix defined by equation (B13), that is, a matrix formed by column vectors Ysj as in Eq. (6.11).

Note also that it follows from Eqs. (6.8) and (6.10) that

detHa=detCa=1na0.
(6.13)

Let us turn now to the eigenvalue problem for the system matrix Ha. In view of Eq. (6.7), an eigenvector Z of Ha satisfies

JnTsDaπnJnsZs=0,Zs=qsps,
(6.14)

or equivalently

JnTsqs+Daps=0,
(6.15)
πnqs+Jnsps=0.
(6.16)

Note first that πnqs0; otherwise, we consequently obtain ps=0 from Eq. (6.16) and then qs=0 from Eq. (6.15), implying Zs=0 contradicting that Zs is an eigenvector. Using that, we normalize qs by the following assumption:

πnqs=enenTqs=s2nen, or equivalently qns=enTqs=s2n.
(6.17)

This particular choice of normalization makes the components of eigenvectors to be polynomials of s rather than rational functions. Combining the explicit formula

Jns1=1s1s21s31n1sn01s1s21n2sn1001s1s20001s
(6.18)

with Eqs. (6.16) and (6.18), we readily obtain

ps=s2nJns1en=1n1sn1nsn+1s2n2s2n1.
(6.19)

Then, plugging expression (6.19) into Eq. (6.15) yields

qs=JnTs1Dap==s2nJnTs1DaJns1en.
(6.20)

Using Eqs. (6.18)–(6.20), we obtain the following expressions for the components of qs and ps:

qjs=1n1k=1jak1snj+2k1,pjs=1n+jsn+j1,1jn.
(6.21)

Consequently, we get the following representation for eigenvector Zs:

HaZs=sZs,Zs=qsps=q1sqnsp1spns=k=02n1Zksk,
(6.22)

where qs and ps are defined by Eq. (6.21).

Note that according to Eqs. (6.21) and (6.22), the eigenvector Zs of the system matrix Ha is uniquely determined by the corresponding eigenvalue s. Evidently, Zs is a vector polynomial of s with vector coefficients Zk, which are determined by expressions (6.21) for vectors qs and ps.

Comparing Eqs. (6.22) and (6.11), we arrive with the following relationship between eigenvectors Zs and Ys:

Zs=TaYs,Ta=Z0|Z1||Z2n1,colTa,k=Zk1,1k2n1.
(6.23)

Note that the 2n × 2n matrix Ta in Eq. (6.11) is defined by its columns, which are the vector coefficients Zk of the vector polynomial Zs. Just as the system matrix Ha and the companion matrix Ca, matrix Ta is completely defined by the system parameters ak and hence by the polynomial χas. An analysis shows that Ta is the 2 × 2 upper triangular block matrix, with blocks of the dimension n × n, and based on that, one can establish that

detTa=a0n.
(6.24)

The significance of matrix Ta is that it provides for the similarity relation between the system matrix Ha and its companion matrix Ca, that is,

Ca=Ta1HaTa.
(6.25)

Equations (7.2), (7.3), (7.6), and .(7.7) show examples of matrices Ha, Ca, and Ta for the cases n = 3, 4.

Note then if we introduce the 2n × 2n matrix

Za=TaYa
(6.26)

and use it in combination with Eq. (6.12), we obtain

Ha=ZaJaZa1,
(6.27)

where Ja is the Jordan form of the companion matrix Ca and hence of the system matrix Ha as well. We refer to Za as the Jordan basis matrix for matrix Ha.

The principal LagrangianLa obtained from the principal Hamiltonian Ha by the Legendre transformation is

La=k=1n1k1ak1vk+1qk+12k=1n1k1ak1vk2+12k=1n1kakqk2+12qn2,vk=tqk,1kn.
(6.28)

An equivalent to the La version of it with the skew-symmetric gyroscopic part is the following Lagrangian:

La=12k=1n1k1ak1vk+1qkvkqk+1+12k=1n1k1ak1vk2+12k=1n1kakqk2+12qn2,vk=tqk,1kn.
(6.29)

The equivalency between two Lagrangians defined by Eqs. (6.28) and (6.29) is understood as that the corresponding EL equations are the same (see Sec. VIII A).

We show in this section the explicit form of matrices Ha, Ca, and Ta related to the principal Hamiltonian defined by Eqs. (5.3) and (5.4). The expressions of these matrices are somewhat different for even and odd n, and with that in mind, we consider two cases of n = 3 and n = 4.

The principal Hamiltonian and the corresponding characteristic polynomials for n = 4 are, respectively,

H=k=13pk+1qk+12k=131k1ak1pk2+12q32,χs=s6a2s4a1s2a0.
(7.1)

The significant matrices in this case are as follows:

H=000a0001000a1001000a2000010000001001000,C=010000001000000100000010000001a00a10a20,
(7.2)
T=00a00000a00a100a00a10a20000100000010000001.
(7.3)

The principal Hamiltonian and the corresponding characteristic polynomials for n = 4 are, respectively,

H=k=14pk+1qk+12k=141k1ak1pk2+12q42,
(7.4)
χs=s8+a3s6+a2s4+a1s2+a0.
(7.5)

The significant matrices in this case are as follows:

H=0000a000010000a100010000a200010000a300000100000000100000000100010000,C=01000000001000000001000000001000000001000000001000000001a00a10a20a30,
(7.6)
T=000a0000000a00a10000a00a10a200a00a10a20a3000001000000001000000001000000001.
(7.7)

We provide here basic facts on the Lagrangian and Hamiltonian structures for linear systems.

The Lagrangian L for a linear system is a quadratic function (bilinear form) of the system state Q=qrr=1n (column vector) and its time derivatives tQ, that is,

L=LQ,tQ=12QtQTMLQtQ,ML=ηθTθα,
(8.1)

where T denotes the matrix transposition operation, and α, η, and θ are n × n-matrices with real-valued entries. In addition to that, we assume matrices α, η to be symmetric, that is,

α=αT,η=ηT.
(8.2)

Consequently,

L=12tQTαtQ+tQTθQ12QTηQ.
(8.3)

Then, by Hamilton’s principle, the system evolution is governed by the EL equations,

ddtLtQLQ=0,
(8.4)

which, in view of Eq. (8.3) for the Lagrangian L, turns into the following second-order vector ordinary differential equation (ODE):

αt2Q+θθTtQ+ηQ=0.
(8.5)

Note that matrix θ enters Eq. (8.5) through its skew-symmetric component 12θθT justifying as a possibility to impose the skew-symmetry assumption on θ, that is,

θT=θ.
(8.6)

Indeed, the symmetric part θs=12θ+θT of the matrix θ is associated with a term to the Lagrangian, which can be recast as is the complete (total) derivative, namely, 12tQTθsQ. It is a well-known fact that adding to a Lagrangian the complete (total) derivative of a function of Q does not alter the EL equations. Namely, the EL equations are invariant under the Lagrangian gauge transform LL+tFq,t [Ref. 24 (2.9 and 2.10) and Ref. 25 (I.2)].

Under assumption (8.6), Eq. (8.5) turns into its version with the skew-symmetric θ,

αt2Q+2θtQ+ηQ=0   if   θT=θ.
(8.7)

It turns out though that our principal Lagrangian that corresponds to the principal Hamiltonian by the Legendre transformation does not have skew-symmetric θ satisfying (8.6). For this reason, we do not impose the condition of skew-symmetry on θ.

The EL equations are the second order ODE. The standard way to reduce them to the equivalent first order ODE yields

tY=LY,Y=QtQ,
(8.8)

where

L=0Iα1ηα1θθT=I00α10IηθθT.
(8.9)

With the spectral analysis of Eq. (8.5) in mind, we can recast it as

AtQ=0,As=αs2+2θs+η,
(8.10)

where evidently As is the n × n-matrix polynomial.

An alternative to Eqs. (8.8) and (8.9) is to replace the second-order vector ODE (8.5) with the first-order one with the Hamilton equations associated with the Hamiltonian H defined by

H=HP,Q=PTtQLQ,tQ,P=LtQ=αtQ+θQ.
(8.11)

Note that the second equation in (8.11) implies the following relations between the velocity and momentum vectors:

tQ=α1PθQ,P=αtQ+θQ.
(8.12)

Consequently,

HP,Q=12PθQTα1PθQ+QTηQ=12tQTαtQ+12QTηQ.
(8.13)

Note also that Eq. (8.12) imply

QP=I0θαQtQ,QtQ=I0α1θα1QP.
(8.14)

H can be interpreted as the system energy, which is a conserved quantity, that is,

tHP,Q=0.
(8.15)

The function HP,Q defined by (8.13) can be recast into the following form:

HP,Q=12QPTMHQP,
(8.16)

where MH is the 2n × 2n matrix having the block form

MH=θTα1θ+ηθTα1α1θα1=IθT0Iη00α1I0θI,
(8.17)

where I is the identity n × n-matrix. The Hamiltonian form of the Euler–Lagrange equation (8.4) reads

tu=JMHu,u=QP,J=0II0.
(8.18)

Matrix J defined in Eq. (8.18) is called the unit imaginary matrix and it satisfies (Ref. 26, 3.1)

J=JT=J1.
(8.19)

Note that in view of Eqs. (8.17) and (8.18), we have

JMH==α1θα1θTα1θηθTα1.
(8.20)

Then, the corresponding to Hamilton vector equation (8.18) matrix similar to the companion polynomial matrix CAs=sBA in (C5) is

CJMH;s=sI00IJMH==s+α1θα1θTα1θ+ηsθTα1.
(8.21)

Let us introduce the matrix

H=JMH.
(8.22)

Note that in view of Eqs. (8.17) and (8.19), we have MHT=MH and

HT=MHJ=JJMHJ=J1HJ,
(8.23)

implying that the transposed to H matrix HT is similar to H.

Note that under the assumption that α−1 exists according to Eqs. (8.1) and (8.17), we have

ML=ηθTθα,MH=ηHθHTθHαH=θTα1θ+ηθTα1α1θα1,
(8.24)

implying

αH=α1,θH=α1θ,ηH=θTα1θ+η
(8.25)

and

α=αH1,θ=αH1θH,η=ηHθHTαH1θH.
(8.26)

If the Lagrangian L depends on higher order derivatives as in

L=12xn2+m=0n1amxm2,xm=tmx,

then the corresponding equations for its extremals are (Ref. 23, 1.2.3 and 3.1.4)

t2nx+m=0n11nmamt2mx=0.

The main point of this section is that in the case when the energy is non-negative, that is, HP,Q0, then the system spectral properties are ultimately determined by a self-adjoint, and hence diagonalizable, operator Ω defined by Eq. (8.33). The argument is as follows:27 Suppose that

α=αT0,η=ηT0.
(8.27)

Then, representations (8.16) and (8.17) combined with the inequalities (8.2) and (8.27) imply

HP,Q0 and MH=MHT0.
(8.28)

Note that matrix MH can be recast as

MH=KTK,
(8.29)

where the matrix K is the block matrix

K=Kq00KpI0θI=Kq0KpθKp,Kq=η,Kp=α1,
(8.30)

which manifestly takes into account the gyroscopic term θ. Here, α and η denote the unique positive semidefinite square roots of the matrices α and η, respectively. In particular, it follows from the properties (8.2) and the proof of Ref. 28, S VI.4, Theorem VI.9 that Kp, Kq are n × n matrices with real-valued entries with the properties

Kp=KpT>0,   Kq=KqT0.
(8.31)

If we introduce now the force variable

v=Ku,
(8.32)

then the evolution equation (8.18) can be recast into the following form:

tv=iΩv,Ω=Ω*=iKJKT=0iΦiΦTΩp,Ωp=i2KpθKpT,Φ=KqKpT,
(8.33)

where Ω is evidently a self-adjoint operator.

Hamiltonian matrices arise naturally as the matrices governing the evolution of Hamiltonian systems (see Sec. VIII B).

Let JR2n×2n be the unit imaginary matrix defined by Eq. (8.18). It satisfies the identities (8.19).

Definition 11
(symplectic matrix). A matrix TR2n×2n is called symplectic if it satisfies the following identity (Ref. 29, 3.1):
TTJT=J,J=0InIn0.
(8.34)
It readily follows from Eqs. (8.34) and (8.19) that the symplectic matrix T is nonsingular and
T1=JTTJ.
It is also evident that T is symplectic if and only if matrices T−1 and TT are symplectic.

Evidently, symplectic matrices in R2n×2n form a group.

Definition 12
(Hamiltonian matrix). A matrix MR2n×2n is called Hamiltonian (or infinitesimally symplectic) if it satisfies the following identity (Ref. 29, 3.1):
J1MTJ=M,J=0InIn0.
(8.35)
Hamiltonian matrix property (8.35) is evidently equivalent to the symmetry of the matrix JM, that is,
JMT=JM.
(8.36)
In other words, a Hamiltonian matrix A is a matrix of the form
A=JA,AT=A.
(8.37)

Since the definition of the Hamiltonian matrix involves a transposed matrix following the general statement, it is of importance to know that a matrix over the field of complex numbers is always similar to its transposed (Ref. 30, 3.2.3).

Proposition 13

(similarity of a matrix and its transposed). LetACn×n. There exists a nonsingular complex symmetric matrixSsuch thatAT = SAS−1.

The following statement provides different equivalent descriptions of a Hamiltonian matrix (Ref. 29, 3.1):

Proposition 14

(Hamiltonian matrix). The following are equivalent: (i)Mis Hamiltonian, (ii)M=JA, whereAis symmetric, and (iii)JAis symmetric. Moreover, ifMandKare Hamiltonian, then so areMT,αM,αR,M ± K, andM,KMKKM.

The following representation holds for a Hamiltonian matrix A (Ref. 26, 3.1):

Proposition 15
(Hamiltonian matrix). A matrixAC2n×2nis a Hamiltonian matrix if and only if there exist matricesA,B,CFn×nsuch thatBandCare symmetric and
A=ABCAT,B=BT,C=CT.
(8.38)
The set of all Hamiltonian matrices forms a Lie algebra.

In fact, a matrix over the field of complex numbers is always similar to its transposed (Ref. 30, 3.2.3).

Proposition 16

(Similarity of a matrix and its transposed). LetACn×n. There exists a nonsingular complex symmetric matrixSsuch thatAT = SAS−1.

The Proof of Proposition 16 can be obtained from the matrix similarity to its Jordan canonical form.

Important spectral properties of Hamiltonian matrices and their canonical forms are studied in Ref. 23 (2.2), Ref. 31, and Ref. 29 (3.3, 4.6, and 4.7). As to the more detailed spectral properties of Hamiltonian matrices, the following statements hold:

Proposition 17

(Jordan structure of a real Hamiltonian matrix). The characteristic polynomial of a real Hamiltonian matrix is an even polynomial. Thus, ifζis an eigenvalue of a Hamiltonian matrix, thenζ,ζ̄, andζ̄are also its eigenvalues with the same multiplicity. The entire Jordan block structure is the same forζ, −ζ,ζ̄, andζ̄.

For the sake of self-consistency, we provide in this section basic information on the basics of the electric network theory and relevant notations.

Electrical networks are a well-established subject represented in many monographs. We present here basic elements of the electrical network theory following mostly to Refs. 32–35. The electrical network theory constructions are based on the graph theory concepts of branches (edges), nodes (vertices), and their incidences. This approach is efficient in loop (fundamental circuit) analysis and the determination of independent variables for the Kirchhoff current and voltage laws—the subjects relevant to our studies here.

We are particularly interested in a conservative electrical network, which is a particular case of an electrical network composed of electric elements of three types: capacitors, inductors, and gyrators. We remind that a capacitor and an inductor are the so-called two-terminal electric elements, whereas a gyrator is a four-terminal electric element as discussed below. We assume that capacitors and inductors can have positive or negative respective capacitances and inductances.

The elementary electric network (circuit) elements of interest here are a capacitor, an inductor, a resistor, and a gyrator [Ref. 32 (1.5 and 2.6) and Ref. 34 (Appendix 5.4)].2,36 These elements are characterized by the relevant voltage–current relationships. These relationships for the capacitor, inductor, and resistor are, respectively, as follows [Ref. 32 (1.5), Ref. 37 (3-Circuit theory), and Ref. 38 (1.3)]:

I=CtV,V=LtI,V=RI,
(9.1)

where I and V are, respectively, the current and the voltage and real C, L, and R are called, respectively, the capacitance, the inductance, and the resistance. The voltage–current relationship for the gyrator depicted in Fig. 7 is

(a):V1V2=GI2GI1,(b):V1V2=GI2GI1,
(9.2)

where I1, I2 and V1, V2 are, respectively, the currents and the voltages, and quantity G is called the gyration resistance.

The common graphic representations of the network elements are depicted in Figs. 6 and 7. The arrow next to the symbol G in Fig. 7 shows the direction of gyration.

FIG. 6.

Capacitance, inductance, and resistance.

FIG. 6.

Capacitance, inductance, and resistance.

Close modal

The gyrator has the so-called inverting property, as shown in Fig. 8 [Ref. 32 (1.5), Ref. 2, Ref. 36, and Ref. 39, (29.1)]. Namely, when a capacitor or an inductor connected to the output port of the gyrator, it behaves as an inductor or capacitor, respectively, with the following effective values:

Lef=G2C,Cef=LG2.
(9.3)
FIG. 8.

(a) Effective inductor; (b) effective capacitor.

FIG. 8.

(a) Effective inductor; (b) effective capacitor.

Close modal

Note that the voltage–current relationships in the second equation in (9.2) can be obtained from the first equation in (9.2) by substituting −G for G. The gyrator is a device that accounts for physical situations in which the reciprocity condition does not hold. The voltage–current relationships in Eq. (9.2) show that the gyrator is a non-reciprocal circuit element. In fact, it is antireciprocal. Note that the gyrator, like the ideal transformer, is characterized by a single parameter G, which is the gyration resistance. The arrows next to the symbol G in Figs. 7(a) and 7(b) show the direction of gyration.

Along with the voltage V and the current I variables, we introduce the charge variable Q and the momentum (per unit of charge) variable P by the following formulas:

Qt=Itdt,It=tQ,
(9.4)
Pt=Vtdt,Vt=tP.
(9.5)

We also introduce the energy stored variable W (Ref. 37, Circuit Theory). Then, the voltage–current relations (9.1) and the stored energy W can be represented as follows:

capacitor: V=QC,I=tQ=CtV,Q=CV=CtP,
(9.6)
W=12VQ=Q22C=CV22=CtP22,
(9.7)
inductor: V=LtI,P=LI=LtQ,tQ=PL,
(9.8)
W=PI2=LI22=LtQ22=P22L,
(9.9)
resistor: V=RI,P=RQ.
(9.10)

The Lagrangian associated with the network elements are as follows:34,40,23,37

capacitor: L=Q22C,inductor: L=LtQ22,
(9.11)
gyrator: L=GQ1tQ2,L=GQ1tQ2Q2tQ12.
(9.12)

Note that the difference between two alternatives for the Lagrangian in Eq. (9.12) is 12GtQ1Q2, which is evidently the complete time derivative. Consequently, the EL equation is the same for both Lagrangians (see Sec. VIII A).

There are a number of physical devices that can provide for negative capacitances and inductances needed for our circuits.7,39 Following Refs. 2 and 36, we show circuits in Fig. 9 that utilize operational amplifiers (called also “opamps”) to achieve negative impedance, capacitance, and inductance, respectively. The currents and voltages for circuits depicted in Fig. 9 are, respectively, as follows: (i) for negative impedance as in Fig. 9(a),

Vin=ZI,Vo=2Vin,I1=I2=VinR,
(9.13)
  • (ii)
    for negative capacitance as in Fig. 9(b),
    Vin=ZinI,Zin=iωC,Vo=2Vin,I1=I2=VinR;
    (9.14)
  • (iii)
    for negative inductance as in Fig. 9(c),
    Vin=ZinI,Zin=iωR2C,I1=I2=VinR,Vo=Vin1+1iωRC.
    (9.15)
FIG. 9.

Operational-amplifier-based negative (a) impedance converter, (b) capacitance converter, and (c) inductance converter.

FIG. 9.

Operational-amplifier-based negative (a) impedance converter, (b) capacitance converter, and (c) inductance converter.

Close modal

An ideal operational amplifier (opamp) assumes the following:2,36

  1. Infinite gain. It means that the opamp has no limit in the amount of voltage it can generate in the output, and the supply voltages can be infinitely large.

  2. Infinitely fast response. It means a large (close to infinite) slew rate.

  3. Infinite bandwidth. It means that the gain of the opamp does not drop at infinitely large frequencies.

  4. Infinite input impedance. It means that the input current to each input port (inverting and non-inverting) is zero. These opamps are voltage controlled devices.

  5. Zero output impedance. It means that the output voltage is not dependent on the load impedance.

Any physical implementation of an opamp has its limitations associated with deviations from above ideal assumptions. The most significant deviation from ideal conditions for opamps is probably their limited frequency band and frequency dependence. Fortunately, the EPD regime is a single frequency phenomenon, and consequently, for that single frequency, proper adjustment of the circuit elements can restore the EPD property. In Sec. IV, we study the most general case of our simplest circuit composed of two LC-loops coupled by a gyrator and derive there the most general conditions under which the relevant evolution matrix exhibits nontrivial Jordan blocks. These most general circuits possessing EPDs provide the maximum flexibility for needed adjustments.

We would like to point out that implementations of opamps as in Fig. 9 are just examples and other implementations with different capabilities are available (see, for instance, Ref. 41 and references therein).

We follow here mostly to Refs. 32 and 33. The purpose of this section is to concisely describe and illustrate relevant concepts with understanding that the precise description of all aspects of the concepts is available in Refs. 32 and 33.

To describe topological (geometric) features of the electric network, we use the concept of linear graph defined as a collection of points, called nodes, and line segments called branches, the nodes being joined together by the branches, as indicated in Fig. 7(b). Branches whose ends fall on a node are said to be incident at the node. For instance, [Fig. 7(b)] branches 1, 2, 3, 4 are incident at node 2. Each branch in Fig. 7(b) carries an arrow indicating its orientation. A graph with oriented branches is called an oriented graph. The elements of a network associated with its graph have both a voltage and a current variable, each with its own reference. In order to relate the orientation of the branches of the graph to these references, the convention is made that the voltage and current of an element have the standard reference—voltage-reference “plus” at the tail of the current-reference arrow. The branch orientation of a graph is assumed to coincide with the associated current reference, as shown in Figs. 6 and 7.

We denote the number of branches of the network by Nb ≥ 2 and the number of nodes by Nn ≥ 2.

A subgraph is a subset of the branches and nodes of a graph. The subgraph is said to be proper if it consists of strictly less than all the branches and nodes of the graph. A path is a particular subgraph consisting of an ordered sequence of branches having the following properties:

  1. At all but two of its nodes, called internal nodes, there are incident exactly two branches of the subgraph.

  2. At each of the remaining two nodes, called the terminal nodes, there is incident exactly one branch of the subgraph.

  3. No proper subgraph of this subgraph, having the same two terminal nodes, has properties 1 and 2.

A graph is called connected if there exists at least one path between any two nodes. We consider here only connected graphs such as shown in Fig. 10(b).

FIG. 10.

The network (a) and its graph (b). There are four nodes marked by small disks (black). In (b), there are three twigs identified by bolder (black) lines and labeled by numbers 1, 3, and 5. There are four links identified by dashed (red) lines and labeled by numbers 2, 4, 6, 7. There are also four oriented f-loops formed by the branches as shown.

FIG. 10.

The network (a) and its graph (b). There are four nodes marked by small disks (black). In (b), there are three twigs identified by bolder (black) lines and labeled by numbers 1, 3, and 5. There are four links identified by dashed (red) lines and labeled by numbers 2, 4, 6, 7. There are also four oriented f-loops formed by the branches as shown.

Close modal

A loop (cycle) is a particular connected subgraph of a graph such that at each of its nodes, there are exactly two incident branches of the subgraph. Consequently, if the two terminal nodes of a path coincide, we get a “closed path”, that is a loop. In Fig. 10(b), branches 7, 1, 3, and 5 together with nodes 1, 2, 3, and 4 form a loop. We can specify a loop by either the ordered list of the relevant branches or the ordered list of the relevant nodes.

We remind that each branch of the network graph is associated with two functions of time t: its current I(t) and its voltage V(t). The set of these functions satisfy two Kirchhoff’s laws [Ref. 32 (2.2), Ref. 33, Ref. 34, Ref. 37 (Circuit Theory), Ref. 35, and Ref. 42]. The Kirchhoff current law (KCL) states that in any electric network, the sum of all currents leaving any node equals zero at any instant of time. The Kirchhoff voltage law (KVL) states that in any electric network, the sum of voltages of all branches forming any loop equals zero at any instant of time. It turns out that the number of independent KCL equations is Nn − 1, and the number KVL equations is Nfl = NbNn + 1 [the first Betti number (Ref. 33, Ref. 34, and Ref. 35, 2.3)].

There is an important concept of a tree in the network graph theory [Ref. 32 (2.2), Ref. 34 (2.1), and Ref. 35 (2.3)]. A tree, known also as a complete tree, is defined as a connected subgraph of a connected graph containing all the nodes of the graph but containing no loops as illustrated in Fig. 10(b). The branches of the tree are called twigs, and those branches that are not on the tree are called links (Ref. 32, 2.2). The links constitute the complement of the tree or the cotree. The decomposition of the graph into a tree and cotree is not unique.

The system of fundamental loops, or system of f-loops for short, [Ref. 32 (2.2), Ref. 34 (2.1), and Ref. 35, (2.3)], is of particular importance to our studies. The system of time-dependent charges (defined as the time integrals of the currents) associated with the system of f-loops provides a c complete set of independent variables. When the network tree is selected, then every link defines the containing it f-loop. The orientation of an f-loop is defined by the orientation of the link it contains. Consequently, there are as many of f-loops as there are in links, and

number of   f-loops:Nfl=NbNn+1.
(9.16)

The number Nfl of f-loops defined by Eq. (9.16) quantifies the connectivity of the network graph, and it is known in the algebraic topology as the first Betti number [Ref. 33, Ref. 34, Ref. 35 (2.3), and Ref. 43.

The discussed concepts of the graph of an electric network such as the tree, twigs, links, and f-loops are illustrated in Fig. 10. In particular, there are four nodes marked by small disks (black). In Fig. 10(b), there are three twigs identified by bolder (black) lines and labeled by numbers 1, 3, 5. There are four links identified by dashed (red) lines and labeled by numbers 2, 4, 6, 7. There are also four oriented f-loops formed by the branches as follows: (1) 7, 1, 3, 5; (2) 2, 1; (3) 4, 3; and (2) 6, 5. These representations of the f-loops as ordered lists of branches identify the corresponding links as the number in the first position in every list.

One also distinguishes simpler planar networks with graphs that can be drawn so that lines representing branches do not intersect. The graph of a general electric network does not have to be planar though. Networks with non-planar graphs can still be represented graphically with more complex display arrangements or algebraically by the incidence matrices (Ref. 32, 2.2).

We developed here a complete mathematical theory allowing for the synthesizing of circuits with evolution matrices exhibiting prescribed Jordan canonical forms subject to natural constraints. In particular, we synthesized simple lossless circuits associated with pairs of Jordan blocks of size 2, 3, and 4, analyzed all their significant properties, and derived closed form algebraic expressions for all significant matrices. Importantly, the elements of the constructed circuits involve negative capacitances and/or inductances. Naturally, those negative values are needed for chosen fixed frequencies only, and that is beneficiary for efficiently achieving them based on operational amplifier converters.

This research was supported by AFOSR Grant No. FA9550-19-1-0103 and Northrop Grumman Grant No. 2326345.

We are grateful to Professor F. Capolino, University of California at Irvine, for reading this manuscript and giving valuable suggestions.

The data that support the findings of this study are available within the article.

We provide here a very concise review of Jordan canonical forms following mostly to Ref. 44 (III.4) and Ref. 30 (3.1 and 3.2). As to a demonstration of how the Jordan block arises in the case of a single nth order differential equation, we refer to Ref. 42, 25.4.

Let A be an n × n matrix and λ be its eigenvalue, and let rλ be the least integer k such that NAλIk=NAλIk+1, where NC is a null space of a matrix C. Then, we refer to Mλ=NAλIrλ as the generalized eigenspace of matrix A corresponding to the eigenvalue λ. Then, the following statements hold (Ref. 44, III.4):

Proposition 18
(generalized eigenspaces). LetAbe ann × nmatrix andλ1, …, λpbe its distinct eigenvalues. Then, generalized eigenspacesMλ1,,Mλpare linearly independent, invariant under the matrixA, and
Cn=Mλ1Mλp.
(A1)
Consequently, any vectorx0inCncan be represented uniquely as
x0=j=1px0,j,x0,jMλj,
(A2)
and
expAtx0=j=1peλjtpjt,
(A3)
where column-vector polynomialspjtsatisfy
pjt=k=0rλj1AλjIktkk!x0,j,x0,jMλj,1jp.
(A4)

For a complex number λ, a Jordan block Jrλ of size r ≥ 1 is an r × r upper triangular matrix of the form

Jrλ=λIr+Kr=λ1000λ1000λ1000λ,J1λ=λ,J2λ=λ10λ,
(A5)
Kr=Jr0=0100001000010000.
(A6)

The special Jordan block Kr=Jr0 defined by Eq. (A6) is a nilpotent matrix that satisfies the following identities:

Kr2=0010000001000000,,Krr1=0001000000000000,Krr=0.
(A7)

A general Jordan n × n matrix J is defined as a direct sum of Jordan blocks, that is,

J=Jn1λ10000Jn2λ20000Jnq1λnq10000Jnqλnq,n1+n2+nq=n,
(A8)

where λj need not be distinct. Any square matrix A is similar to a Jordan matrix as in Eq. (A8), which is called the Jordan canonical form of A. Namely, the following statement holds (Ref. 30, 3.1):

Proposition 19
(Jordan canonical form). LetAbe ann × nmatrix. Then, there exists a non-singularn × nmatrixQsuch that the following block-diagonal representation holds:
Q1AQ=J,
(A9)
whereJis the Jordan matrix defined byEq. (A8)andλj, 1 ≤ jq, are not necessarily different eigenvalues of matrixA. Representation(A9)is known as the Jordan canonical form of matrixA, and matricesJjare called Jordan blocks. The columns of then × nmatrixQconstitute the Jordan basis, providing for the Jordan canonical form(A9)of matrixA.

A function fJrs of a Jordan block Jrs is represented by the following equation [Ref. 45 (7.9) and Ref. 26 (10.5)]:

fJrs=fsfs2fs2r1fsr1!0fsfsr2fsr2!00fsfs000fs.
(A10)

Note that any function fJrs of the Jordan block Jrs is evidently an upper triangular Toeplitz matrix.

There are two particular cases of formula (A10), which can be also derived straightforwardly using Eq. (A7),

expKrt=k=0r1tkk!Krk=1tt22!tr1r1!01ttr2r2!001t0001,
(A11)
Jrs1=k=0r1sk1Krk=1s1s21s31r1sr01s1s21r2sr1001s1s20001s.
(A12)

The companion matrix Ca for the monic polynomial

as=sν+1kνaνksνk,
(B1)

where coefficients ak are complex numbers, is defined by (Ref. 26, 5.2)

Ca=0100001000001a0a1aν2aν1.
(B2)

Note that

detCa=1νa0.
(B3)

An eigenvalue is called cyclic (nonderogatory) if its geometric multiplicity is 1. A square matrix is called cyclic (nonderogatory) if all its eigenvalues are cyclic (Ref. 26, 5.5). The following statement provides different equivalent descriptions of a cyclic matrix (Ref. 26, 5.5):

Proposition 20
(criteria for a matrix to be cyclic). LetACn×nbe ann × nmatrix with complex-valued entries. LetspecA=ζ1,ζ2,,ζrbe the set of all distinct eigenvalues, andkj=indAζjis the largest size of the Jordan block associated withζj. Then, the minimal polynomialμAsof the matrixA, that is, a monic polynomial of the smallest degree such thatμAA=0, satisfies
μAs=j=1rsζjkj.
(B4)
Furthermore, the following statements are equivalent:

  1. μAs=χAs=detsIA.

  2. A is cyclic.

  3. For every ζj, the Jordan form of A contains exactly one block associated with ζj.

  4. A is similar to the companion matrix CχA.

Proposition 21
(companion matrix factorization). Letasbe a monic polynomial having degreeνandCais itsν × νcompanion matrix. Then, there exist unimodularν × νmatricesS1sandS2s, that is,detSm=±1,m = 1, 2, such that
sIνCa=S1sIν10ν1×101×ν1asS2s.
(B5)
Consequently,Cais cyclic and
χCas=μCas=as.
(B6)

The following statement summarizes important information on the Jordan form of the companion matrix and the generalized Vandermonde matrix [Ref. 26 (5.16), Ref. 46 (2.11), and Ref. 45 (7.9)].

Proposition 22
(Jordan form of the companion matrix). LetCabe ann × na companion matrix of the monic polynomialasdefined by equation(B1). Suppose that the set of distinct roots of polynomialasisζ1,ζ2,,ζrandn1,n2,,nris the corresponding set of the root multiplicities such that
n1+n2++nr=n.
(B7)
Then,
Ca=RJR1,
(B8)
where
J=diagJn1ζ1,Jn2ζ2,,Jnrζr
(B9)
is the Jordan form of the companion matrixCaand then × nmatrixRis the so-called generalized Vandermonde matrix defined by
R=R1|R2||Rr,
(B10)
whereRjis ann × njmatrix of the form
Rj=100ζj10ζjn2n21ζjn3n2nj1ζjnnj1ζjn1n11ζjn2n1nj1ζjnnj.
(B11)
As a consequence of representation(B9),Cais a cyclic matrix.

As to the structure of matrix Rj in equation (B11), if we denote by Yζj its first column, then it can be expressed as follows (Ref. 46, 2.11):

Rj=Y0|Y1||Ynj1,Ym=1m!sjmYζj,0mnj1.
(B12)

In the case when all eigenvalues of a cyclic matrix are distinct, then the generalized Vandermonde matrix turns into the standard Vandermonde matrix

V=111ζ1ζ2ζnζ1n2ζ2n2ζnn2ζ1n1ζ2n1ζnn1.
(B13)

An important incentive for considering matrix polynomials is that they are relevant to the spectral theory of the differential equations of the order higher than 1, particularly the Euler–Lagrange equations that are the second-order differential equations in time. We provide here selected elements of the theory of matrix polynomials following mostly to Ref. 47 (II.7 and II.8) and Refs. 34 and 48. The general matrix polynomial eigenvalue problem reads

Asx=0,As=j=0νAjsj,x0,
(C1)

where s is complex number, Ak are constant m × m matrices, and xCm is an m-dimensional column-vector. We refer to problem (C1) of funding complex-valued s and non-zero vector xCm as the polynomial eigenvalue problem.

If a pair of a complex s and non-zero vector x solves problem (C1), we refer to s as an eigenvalue or as a characteristic value and to x as the corresponding to seigenvector. Evidently, the characteristic values of problem (C1) can be found from the polynomial characteristic equation

detAs=0.
(C2)

We refer to the matrix polynomial As as regular if detAs is not identically zero. We denote by ms0 the multiplicity (called also algebraic multiplicity) of the eigenvalue s0 as a root of the polynomial detAs. In contrast, the geometric multiplicity of the eigenvalue s0 is defined as dimkerAs0, where kerA defined for any square matrix A stands for the subspace of solutions x to the equation Ax = 0. Evidently, the geometric multiplicity of the eigenvalue does not exceed its algebraic one (see Corollary 25).

It turns out that the matrix polynomial eigenvalue problem (C1) can be always recast as the standard “linear” eigenvalue problem, namely,

sBAx=0,
(C3)

where × matrices A and B are defined by

B=I0000I0000I0000Aν,A=0I0000I00000IA0A1Aν2Aν1,
(C4)

with I being the m × m identity matrix. Matrix A, particularly in the monic case, is often referred to as the companion matrix. In the case of the monic polynomialAλ, when Aν=I is an m × m identity matrix, matrix B = I is an × identity matrix. The reduction of the original polynomial problem (C1) to an equivalent linear problem (C3) is called linearization.

The linearization is not unique, and one way to accomplish is by introducing the so-called known “companion polynomial,” which is an × matrix

CAs=sBA=sII000sII000sIIA0A1Aν2sAν+Aν1.
(C5)

Note that in the case of the EL equations, the linearization can be accomplished by the relevant Hamilton equations.

To demonstrate the equivalency between the eigenvalue problems for the × companion polynomial CAs and the original m × m matrix polynomial As, we introduce two × matrix polynomials Es and Fs. Namely,

Es=E1sE2sEν1sII0000I0000I0,
(C6)
detEs=1,

where m × m matrix polynomials Ejs are defined by the following recursive formulas:

Eνs=Aν,Ej1s=Aj1+sEjs,j=ν,,2.
(C7)

The matrix polynomial Fs is defined by

Fs=I000sII000sII000sII,detFs=1.
(C8)

Note that both matrix polynomials Es and Fs have constant determinants, readily implying that their inverses E1s and F1s are also matrix polynomials. Then, it is straightforward to verify that

EsCAsF1s=EssBAF1s=As0000I0000I0000I.
(C9)

Identity (C9) where matrix polynomials Es and Fs have constant determinants can be viewed as the definition of equivalency between the matrix polynomial As and its companion polynomial CAs.

Let us take a look at the eigenvalue problem for the eigenvalue s and eigenvector xCmν associated with the companion polynomial CAs, that is,

sBAx=0,x=x0x1x2xν1Cmν,xjCm,0jν1,
(C10)

where

sBAx=sx0x1sx1x2sxν2xν1j=0ν2Ajxj+sAν+Aν1xν1.
(C11)

With equations (C10) and (C11) in mind, we introduce the following vector polynomial:

xs=x0sx0sν2x0sν1x0,x0Cm.
(C12)

Not accidentally, the components of the vector xs in its representation (C12) are in evident relation with the derivatives tjx0est=sjx0est. That is just another sign of the intimate relations between the matrix polynomial theory and the theory of systems of ordinary differential equations (see  Appendix D).

Theorem 23
(eigenvectors). LetAsas in equations(C1)be regular, that is, detAsis not identically zero, and let × matricesAandBbe defined by equations(C2). Then, the following identities hold:
sBAxs=000Asx0,xs=x0sx0sν2x0sν1x0,
(C13)
detAs=detsBA,detB=detAν,
(C14)
wheredetAs=detsBAis a polynomial of degreeifdetB=detAν0. There is one-to-one correspondence between solutions of equationsAsx=0andsBAx=0. Namely, a pairs, xsolves the eigenvalue problemsBAx=0if and only if the following equalities hold:
x=xs=x0sx0sν2x0sν1x0,Asx0=0,x00;detAs=0.
(C15)

Proof.

Polynomial vector identity (C13) readily follows from equations (C11) and (C12). Identities (C14) for the determinants follow straightforwardly from equations (C12), (C15), and (C9). If detB=detAν0, then the degree of the polynomial detsBA has to be since A and B are × matrices.

Suppose that equations (C15) hold. Then, combining them with the proven identity (C13), we get sBAxs=0, proving that expressions (C15) define an eigenvalue s and an eigenvector x = xs.

Suppose now that sBAx=0, where x≠0. Combining that with equations (C11), we obtain
x1=sx0,x2=sx1=s2x0,,xν1=sν1x0,
(C16)
implying that
x=xs=x0sx0sν2x0sν1x0,x00,
(C17)
and
j=0ν2Ajxj+sAν+Aν1xν1=Asx0.
(C18)
Using equations (C17) and identity (C13), we obtain
0=sBAx=sBAxs=000Asx0.
(C19)
Equation (C19) readily imply Asx0=0 and detAs=0 since x0 ≠ 0. This completes the proof.□

Remark 24

(characteristic polynomial degree). Note that according to Theorem 23, the characteristic polynomial detAs for the m × m matrix polynomial As has degree , whereas in the linear case sIA0 for m × m identity matrix I and m × m matrix A0, the characteristic polynomial detsIA0 is of degree m. This can be explained by observing that in the non-linear case of the m × m matrix polynomial As, we are dealing effectively with many more m × m matrices A than just a single matrix A0.

Another problem of our particular interest related to the theory of matrix polynomials is eigenvalue and eigenvector degeneracy and consequently the existence of non-trivial Jordan blocks, that is, Jordan blocks of dimensions higher or equal to 2. The general theory addresses this problem by introducing the so-called “Jordan chains,” which are intimately related to the theory of system of differential equations expressed as Atxt=0 and their solutions of the form xt=ptest, where pt is a vector polynomial [see  Appendix D, Ref. 47 (I and II), and Refs. 34 and 48]. Avoiding the details of Jordan chain developments, we simply note that an important point of Theorem 23 is that there is one-to-one correspondence between solutions of equations Asx=0 and sBAx=0, and it has the following immediate implication:

Corollary 25
(equality of the dimensions of eigenspaces). Under the conditions of Theorem 23 for any eigenvalues0, that is,detAs0=0, we have
dimkers0BA=dimkerAs0.
(C20)
In other words, the geometric multiplicities of the eigenvalues0associated with matricesAs0ands0BAare equal. In view of identity(C20), the following inequality holds for the (algebraic) multiplicityms0:
ms0dimkerAs0.
(C21)

The next statement shows that if the geometric multiplicity of an eigenvalue is strictly less than its algebraic one, then there exist non-trivial Jordan blocks, that is, Jordan blocks of dimensions higher or equal to 2.

Theorem 26
(non-trivial Jordan block). Assuming notations introduced in Theorem 23, let us suppose that the multiplicityms0of eigenvalues0satisfies
ms0>dimkerAs0.
(C22)
Then, the Jordan canonical form of the companion polynomialCAs=sBAhas a least one nontrivial Jordan block of the dimension exceeding 2.
In particular, if
dimkers0BA=dimkerAs0=1
(C23)
andms02, then the Jordan canonical form of companion polynomialCAs=sBAhas exactly one Jordan block associated with eigenvalues0and its dimension isms0.

The Proof of Theorem 26 follows straightforwardly from the definition of the Jordan canonical form and its basic properties. Note that if equations (C23) hold, this implies that the eigenvalue 0 is cyclic (nonderogatory) for matrix As0 and the eigenvalue s0 is cyclic (nonderogatory) for matrix B−1A, provided B−1 exists (see  Appendix B).

In this section, we relate the vector ordinary equations to the matrix polynomials reviewed in  Appendix C, following Ref. 49 (5.1 and 5.7), Ref. 47 (II.8.3), Ref. 44 (III.4), and Ref. 45 (7.9)].

The equation Asx=0 with the polynomial matrix As defined by equations (C1) corresponds to the following m-vector ν-th order ordinary differential:

Atxt=0, where At=j=0νAjtj,
(D1)

where Aj are m × m matrices. Then, differential equation (D1) can be recast in a standard fashion as the -vector first order differential equation

BtYt=AYt,
(D2)

where A and B are × companion matrices defined by equations (C4) and

Yt=xttxttν2xttν1xt
(D3)

is the -column-vector function.

In the case when Aν is an invertible m × m matrix, Eq. (D2) can be recast further as

tYt=ȦYt,
(D4)

where

Ȧ=0I0000I00000IȦ0Ȧ1Ȧν2Ȧν1,Ȧj=Aν1Aj,0ν1.
(D5)

Note that one can interpret Eq. (D4) as a particular case of Eq. (D2) where matrices Aν and B are identity matrices of the respective dimensions m × m and × and that the polynomial matrix As defined by Eq. (C1) becomes the monic matrix polynomial Ȧs, that is,

Ȧs=Isν+j=0ν1Ȧjsj,Ȧj=Aν1Aj,0ν1.
(D6)

Note that in view of Eq. (D3), one recovers xt from Yt by the following formula: