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.

## I. INTRODUCTION

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 systems^{6–10} 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, $H=SJS\u22121$, where *S* is an invertible matrix and $J$ is a block diagonal matrix of the form

*ζ*_{j} are real or complex numbers, and $Jn\zeta $ is the so-called Jordan block, which is an *n* × *n* matrix. For *n* = 1, the matrix $Jn\zeta =\zeta $ turns just into number *ζ*.

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

where *X* is the 2*n* dimensional vector-column describing the circuit state and $H$ is the 2*n* × 2*n* 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 2*n* × 2*n* 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 *n**fundamental 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 $\u2202tqkt$. 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

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 $\u2212HT$, 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 *M*^{T} 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*, $s\u0304$, and $\u2212s\u0304$, where $s\u0304$ is complex-conjugate to *s*, are its eigenvalues as well; (ii) the Jordan blocks corresponding to *s*, −*s*, $s\u0304$, and $\u2212s\u0304$ 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 *J*_{M} is completely determined by its *characteristic polynomia*l $\chi s=detsI\u2212M$, where $I$ is the identity matrix of the relevant dimension. Namely, every eigenvalue *s*_{0} of *M* of multiplicity *m* is associated with the single Jordan block $Jms0$ in the Jordan form *J*_{M} of *M*. Consequently, *for a cyclic matrix* *M*, *its characteristic polynomial* $\chi s=detsI\u2212M$ *encodes all the information about its Jordan form* *J*_{M}. Another property of any cyclic matrix *M* associated with the monic polynomial *χ* is that it is similar to the so-called companion matrix *C*_{χ} defined by the simple explicit expression involving the coefficients of the polynomial *χ* (see Appendix B and references therein). The companion matrix *C*_{χ} is naturally related to the high-order differential equation $\chi \u2202txt=0$, 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 2*n* × 2*n* 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 $\chi s=detsI2n\u2212J$, which is an even monic polynomial $\chi s$ of degree 2*n*. We then consider the companion to $\chi s$ matrix $C$ (see Appendix B), which by the design has $J$ as its Jordan form, that is,

where the columns of matrix $Y$ form the so-called Jordan basis of the companion matrix $C$ associated with the characteristic polynomial $\chi s=detsI2n\u2212J$ (see Appendix B). We proceed with an introduction of our principal Hamiltonian $H$ (see Sec. V) and recover from it the 2*n* × 2*n* 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 2*n* × 2*n* matrix *T* such that

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 prescribed$J$ 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 $\chi s$ 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.

## II. PRINCIPAL CIRCUIT

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 *L*_{j}, *C*_{j}, and *G*_{j} 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:

where *ω*_{0} > 0 is a unit of frequency and 1 ≤ *j* ≤ *n* and $L\u0306$ is the scaled Lagrangian. To have less cluttered formulas, we actually omit “hat” from $L\u0306$, $t\u0303$, $C\u0306j$, $L\u0306j$, and $G\u0306j$ and simply remember from now on that we use the relevant letters for the dimensionless quantities and the scaled Lagrangian.

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

where *q*_{k} and *i*_{k} 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

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 *∂*_{t}*q*_{k} there. Indeed, according to the gyrator settings, the *k*th gyrator has exactly two incoming currents *∂*_{t}*q*_{k} and *∂*_{t}*q*_{k+1}. Indeed, the outgoing current, which passes through the gyrator branch common to the *k*th and the $k+1$-th f-loops, is equal to the sum *∂*_{t}*q*_{k} + *∂*_{t}*q*_{k+1}. When exiting this common branch, the current *∂*_{t}*q*_{k} + *∂*_{t}*q*_{k+1} splits into currents *∂*_{t}*q*_{k} and *∂*_{t}*q*_{k+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 $\chi s$ that has to be of the form

where parameters *a*_{k} are real-valued and satisfy

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 $\chi s$ has to be even polynomial as indicated by Eq. (2.5), and its parameters *a*_{k} 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 $\chi s$ by the setting up the following expressions for the circuit electric inductances, capacitances, and gyration resistances in terms of the coefficients *a*_{k} of the polynomial $\chi s$:

as well as following expressions for coefficients *a*_{j} in terms of the circuit parameters:

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 $L\u2032$ 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

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

### A. Principle circuit for two loops

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

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

and the corresponding EL equations are

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

k | 1 | 2 |

L_{k} | $1a0$ | $\u22121a1$ |

C_{k} | a_{1} | −1 |

G_{k} | $1a1$ |

k | 1 | 2 |

L_{k} | $1a0$ | $\u22121a1$ |

C_{k} | a_{1} | −1 |

G_{k} | $1a1$ |

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:

The determinants of the above matrices are as follows:

The similarity between matrices $H$ and $C$ takes here the form

and can be verified by showing that $HT=CT\u22121$ 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:

## III. SPECIAL CIRCUITS

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

for real *a* and *b* corresponding to the Jordan form $J$ made of two Jordan blocks $Jn\xb1a$ and $Jn\xb1bi$, 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 $\chi s$ defined by Eq. (3.1) for arbitrary *n* ≥ 2.

We are also interested in

where *a*≠0 and *b* ≠ 0 corresponding to the Jordan form $J$ made of four Jordan blocks $J2\xb1a\xb1bi$ and $J2\xb1a\u2213bi$.

### A. Special circuits for two real or pure imaginary eigenvalues and *n* = 2

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

and assign to the circuit elements the values provided in Table II [the parameter *r* refers to the roots of the characteristic polynomial $\chi s$, that is, $\chi r=0$].

k | 1 | 2 |

L_{k} | $1a4$ | $12a2$ |

C_{k} | −2 a^{2} | −1 |

G_{k} | $\u221212a2$ | |

r; n | ±a; 2 | |

k | 1 | 2 |

L_{k} | $1b4$ | $\u221212b2$ |

C_{k} | 2 b^{2} | −1 |

G_{k} | $12b2$ | |

r; n | ±bi; 2 |

k | 1 | 2 |

L_{k} | $1a4$ | $12a2$ |

C_{k} | −2 a^{2} | −1 |

G_{k} | $\u221212a2$ | |

r; n | ±a; 2 | |

k | 1 | 2 |

L_{k} | $1b4$ | $\u221212b2$ |

C_{k} | 2 b^{2} | −1 |

G_{k} | $12b2$ | |

r; n | ±bi; 2 |

In the case of the first polynomial $\chi s$ that has real roots ±*a*, the circuit significant matrices are as follows:

In the case of the second polynomial $\chi s$ that has pure imaginary roots ±*b*i, the significant circuit matrices are

Applying to the case *n* = 2 the general formulas (3.34) for the eigenfrequencies $\omega j=\xb11LjCj$ of *LC*-loops as they were decoupled, we obtain

### B. Special circuit for two real or pure imaginary eigenvalues and *n* = 3

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 $\chi s$, that is, $\chi r=0$]. The corresponding polynomial is

In the case when the roots of $\chi s$ are real numbers ±*a*, the circuit matrices are

k | 1 | 2 | 3 |

L_{k} | $1a6$ | $13a4$ | $13a2$ |

C_{k} | −3 a^{4} | −3 a^{2} | −1 |

G_{k} | $\u221213a4$ | $\u221213a2$ | |

r; n | ±a; 3 | ||

k | 1 | 2 | 3 |

L_{k} | $\u22121b6$ | $13b4$ | $\u221213b2$ |

C_{k} | −3 b^{4} | 3 b^{2} | −1 |

G_{k} | $\u221213b4$ | $13b2$ | |

r; n | ±bi; 3 |

k | 1 | 2 | 3 |

L_{k} | $1a6$ | $13a4$ | $13a2$ |

C_{k} | −3 a^{4} | −3 a^{2} | −1 |

G_{k} | $\u221213a4$ | $\u221213a2$ | |

r; n | ±a; 3 | ||

k | 1 | 2 | 3 |

L_{k} | $\u22121b6$ | $13b4$ | $\u221213b2$ |

C_{k} | −3 b^{4} | 3 b^{2} | −1 |

G_{k} | $\u221213b4$ | $13b2$ | |

r; n | ±bi; 3 |

In the case when the roots of $\chi s$ are pure imaginary numbers ±*b*i, the circuit matrices are

Applying to the case *n* = 3 the general formulas (3.34) for the eigenfrequencies $\omega j=\xb11LjCj$ of *LC*-loops as they were decoupled, we obtain

### C. Special circuit for two real or pure imaginary eigenvalues and *n* = 4

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.

k | 1 | 2 | 3 | 4 |

L_{k} | $1a8$ | $14a6$ | $16a4$ | $14a2$ |

C_{k} | −4 a^{6} | −6 a^{4} | −4 a^{2} | −1 |

G_{k} | $\u221214a6$ | $\u221216a4$ | $\u221214a2$ | |

r; n | ±a; 4 |

k | 1 | 2 | 3 | 4 |

L_{k} | $1a8$ | $14a6$ | $16a4$ | $14a2$ |

C_{k} | −4 a^{6} | −6 a^{4} | −4 a^{2} | −1 |

G_{k} | $\u221214a6$ | $\u221216a4$ | $\u221214a2$ | |

r; n | ±a; 4 |

k | 1 | 2 | 3 | 4 |

L_{k} | $1b8$ | $\u221214b6$ | $16b4$ | $\u221214b2$ |

C_{k} | 4 b^{6} | −6 b^{4} | 4 b^{2} | −1 |

G_{k} | $14b6$ | $\u221216b4$ | $14b2$ | |

r; n | ±bi; 4 |

k | 1 | 2 | 3 | 4 |

L_{k} | $1b8$ | $\u221214b6$ | $16b4$ | $\u221214b2$ |

C_{k} | 4 b^{6} | −6 b^{4} | 4 b^{2} | −1 |

G_{k} | $14b6$ | $\u221216b4$ | $14b2$ | |

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 $\chi s$, that is, $\chi r=0$].

The corresponding polynomial is

In the case when the roots of $\chi s$ are real numbers ±*a*, the circuit matrices are

In the case when the roots of $\chi s$ are pure imaginary numbers ±*b*i, the circuit matrices are

Applying to the case *n* = 4 the general formulas (3.34) for the eigenfrequencies $\omega j=\xb11LjCj$ of *LC*-loops as they were decoupled, we obtain

### D. Special circuit for Hamiltonian quadruple of complex eigenvalues and *n* = 2

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

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 $\chi s$, that is, $\chi r=0$]. 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.

k | 1 | 2 | 3 | 4 |

L_{k} | $1a2+b24$ | $\u221214a2\u2212b2a2+b22$ | $123a4+3b4\u2212a2b2$ | $\u221214a2\u2212b2$ |

C_{k} | $4a2\u2212b2a2+b22$ | $\u221223a4+3b4\u2212a2b2$ | $4a2\u2212b2$ | −1 |

G_{k} | $14a2\u2212b2a2+b22$ | $\u2212122a2b2\u22123a4\u22123b4$ | $14a2\u2212b2$ | |

r; n | a ± bi, −a ± bi; 2 |

k | 1 | 2 | 3 | 4 |

L_{k} | $1a2+b24$ | $\u221214a2\u2212b2a2+b22$ | $123a4+3b4\u2212a2b2$ | $\u221214a2\u2212b2$ |

C_{k} | $4a2\u2212b2a2+b22$ | $\u221223a4+3b4\u2212a2b2$ | $4a2\u2212b2$ | −1 |

G_{k} | $14a2\u2212b2a2+b22$ | $\u2212122a2b2\u22123a4\u22123b4$ | $14a2\u2212b2$ | |

r; n | a ± bi, −a ± bi; 2 |

### E. Special circuits for two real or pure imaginary eigenvalues for *n* ≥ 2

We consider here the special circuits associated with the characteristic polynomials $\chi s$ defined by Eq. (3.1) for arbitrary *n* ≥ 2. Namely, in the case of $\chi s=s2\u2212a2n$ using the binomial formula and expression (2.5) for the general characteristic polynomial, we obtain the following expressions for coefficients:

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:

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

The eigenfrequencies *ω*_{j} in Eq. (3.28) are evidently pure imaginary.

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

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

## IV. JORDAN FORM OF A CIRCUIT COMPOSED OF TWO *LC*-LOOPS AND A GYRATOR

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 *L*_{1}, *C*_{1}, *L*_{2}, *C*_{2}, and *G*_{1} 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:

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

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

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

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

The characteristic polynomial associated with the matrix polynomial $As$ and its linearized version $sI\u2212C$ is

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

Our primary goal here is to identify all real non-zero values of the circuit parameters *L*_{1}, *C*_{1}, *L*_{2}, *C*_{2}, and *G*_{1} 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:

*Let*

*s*

_{0}

*be an eigenvalue of the companion matrix*$C$

*defined by*

*Eq. (4.4)*

*such that its algebraic multiplicity*$ms0\u22652$

*. Then (i)*

*s*

_{0}≠ 0

*; (ii)*−

*s*

_{0}

*is also an eigenvalue of*$C$

*; (iii)*

*s*

_{0}

*is either real or pure imaginary; (iv)*$ms0=m\u2212s0=2$

*; and (v) the Jordan form*$J$

*of the matrix*$C$

*is*

*That is, because of the special form the of companion matrix*$C$,

*the eigenvalue degeneracy for*$C$

*implies that its Jordan form*$J$

*consists of two Jordan blocks of size*2.

The eigenvalue *s*_{0} satisfies $\chi s0=0$. Since in view of Eq. (4.6), $\chi 0=1L1C1L2C2\u22600$, we infer that *s*_{0} ≠ 0. The characteristic equation (4.7) implies that $\chi \u2212s0=\chi s0=0$, and hence, −*s*_{0} is an eigenvalue. Note that since all coefficients of the characteristic equation (4.7) are real, then the number $s\u03040$, which is complex-conjugate to *s*_{0}, is also an eigenvalue since $\chi s\u03040=\chi s0\u0304=0$. If *s*_{0} would be a complex number with non-zero real and imaginary parts, then there would be four distinct eigenvalues *s*_{0}, −*s*_{0}, $s\u03040$, and $\u2212s\u03040$ for the fourth-degree characteristic equation. This would make it impossible for the algebraic multiplicity of *s*_{0} to satisfy $ms0\u22652$, which is a condition of the theorem. Hence, we have to infer that *s*_{0} is either real or pure imaginary. Consequently, *s*_{0} and −*s*_{0} are the only eigenvalues and are the roots of the characteristic polynomial $\chi s$. Since $\chi s$ involves only even degrees of *s*, we also have $ms0=m\u2212s0$, implying $ms0=m\u2212s0=2$.

*s*= 0 and then $1L1C1=0$, but the later is impossible. Note also

*s*

_{0}of the size $ms0=2$, and the same statement holds for −

*s*

_{0}. Consequently, the Jordan form of matrix $C$ satisfies Eq. (4.8), and this completes the proof of the theorem.□

### A. Characteristic equation and eigenvalue degeneracy

The further analytical developments suggest introducing the following variables:

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

Being interested in degenerate eigenvalues satisfying the equation $\chi s=\chi h=0$, we turn to the discriminant Δ_{h} of the quadratic polynomial *χ*_{h} defined by Eq. (4.13), namely,

Recall that the solutions to the quadratic equation *χ*_{h} = 0 are

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

where *h*_{±} satisfy Eq. (4.15).

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

Equation (4.17) is evidently a quadratic equation for *g*. Being given remaining circuit coefficients *ξ*_{1}, *ξ*_{2}, *L*_{1}, and *L*_{2}, this quadratic in the *g* equation has exactly two solutions,

We refer to *g*_{δ} in Eq. (4.18) as *special values of the gyration parameter**g*. For the two special values $g\u0307$, we get from Eq. (4.15) the corresponding two degenerate roots

Since *G*_{1} is real, then $g=G12$ is real as well. Expression (4.18) for *g* is real-valued if and only if

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 $sign\xi 1=sign\xi 2$ is a necessary condition for the eigenvalue degeneracy condition Δ_{h} = 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 *g*_{δ} can be recast as

where $\xi >0$ 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)

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

*Note that the sign constraints**(4.23)**involving the circuit index**σ**defined by**(4.20)**are necessary for the eigenvalue degeneracy condition* Δ_{h} = 0. Combining Eqs. (4.21) and (4.22), we obtain

Since $g=G12>0$, the special values of the gyrator resistance $G\u03071$ corresponding to the special values *g*_{δ} as in Eq. (4.24) are

where the binary variable *σ*_{1} takes values ±1.

Using representation (4.24) for *g*_{δ} under the circuit sign constraints (4.23), we can recast the expression for the degenerate root $h\u0307$ in Eq. (4.19) as follows:

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

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

(the signs of the circuit parameters). *Let the circuit be as depicted in* Fig. 2 *and all its parameters* *L*_{1}*,* *C*_{1}*,* *L*_{2}*,* *C*_{2}, *and* *G*_{1} *be real and non-zero. Then, the companion matrix* $C$ *satisfying equations (4.4) and (4.12)* *has a degenerate eigenvalue if and only if (i) the sign constraints* *(4.23)* *hold and (ii)* $g=G12$ *satisfies* *Eq. (4.24)**. Under the circuit sign constraints* *(4.23)*, *the degenerate solution* *h*_{δ} *to the quadratic equation* *χ*_{h} = 0 *[see* *Eq. (4.13)**] is determined by* *Eq. (4.26)**.*

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.

*Let the circuit be as depicted in*Fig. 2,

*and let all its parameters*

*L*

_{1}

*,*

*C*

_{1}

*,*

*L*

_{2}

*,*

*C*

_{2},

*and*

*G*

_{1}

*be real and non-zero. Then, the companion matrix*$C$

*satisfying (4.4) and (4.12) has the Jordan form*

*if and only if the circuit parameters satisfy the degeneracy conditions described in Lemma 2. Then, for*

*g*=

*g*

_{δ},

*we have*

*where*$\xi >0$

*for*

*ξ*> 0

*,*

*δ*= ±1,

*and*

*σ*

*is the circuit sign index defined by*

*Eq. (4.20)*

*. According to formula*

*(4.30)*,

*degenerate eigenvalues*±

*s*

_{0}

*depend on the product*

*σδ*

*and*$\xi 1,\xi 2$,

*and consequently, they are either real or pure imaginary depending on whether*

*δ*=

*σ*

*or*

*δ*= −

*σ*

*.*

*Note that in the special case when*$\xi 1=\xi 2$,

*the parameter*

*g*

_{δ}

*takes only one non-zero value, namely,*

*whereas*

*g*

_{−1}= 0,

*which is inconsistent with our assumption*

*G*

_{1}≠ 0

*. Evidently, for*

*g*= 0,

*the circuit breaks into two independent*

*LC*

*-circuits, and in this case, the relevant Jordan form is a diagonal*4 × 4

*matrix with eigenvalues*$\xb1\xi 1$

*and*$\xb1\xi 2$

*.*

(instability and marginal stability)*.* Note that according to formula (4.30), degenerate eigenvalues ±*s*_{0} 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 ±*s*_{0} 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:

and this corresponds to

It follows then from representation (4.24) that the corresponding special values *g*_{δ} are

To explain the rise of the circular part of the set *S*_{eig} in Fig. 5, we recast the characteristic equation (4.13) as follows:

Note that

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

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

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 $H,H\u22121$.

Since the eigenvalues *s* satisfy $s=\xb1h$, the established above properties of $h=\xi 1\xi 2H$ can recast for *s* as follows:

*For every*

*g*> 0,

*every solution*

*s*

*to 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:*

*where*

*s*

*is a solution to the characteristic equation (4.7)*

*. Then, for*

*g*

_{−1}≤

*g*≤

*g*

_{1},

*the quadruple of solutions belongs to the circle*$s=\xi 1\xi 24$

*such that*

*where*

*R*

*is defined in relations*

*(4.35)*

*. If*

*g*<

*g*

_{−1}

*or*

*g*>

*g*

_{1},

*the quadruple of solutions consists of either real numbers or pure imaginary numbers if*

*R*> 2

*or*

*R*< −2,

*respectively. In view of relations*

*(4.35)*

*and*$s=\xb1h$,

*where*$h=\xi 1\xi 2H$,

*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 $s=\xi 1\xi 24$ 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 *g*_{1} at which the eigenvalues degenerate.

(transition at degeneracy points)*.* According to formula (4.30), there is a total of four degenerate eigenvalues ±*s*_{0}, namely, $\xb1\xi 1\xi 24$ and $\xb1i\xi 1\xi 24$ [depicted as solid diamond (blue) dots in Fig. 5], that are associated with the two special values of the gyration parameter $g\xb11=\xi 1\xb1\xi 22L1L2$. 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* < *g*_{1} and *g* gets close to either *g*_{−1} or *g*_{1}, the corresponding four distinct eigenvalues on the circle $s=\xi 1\xi 24$ get close to either $\xb1\xi 1\xi 24$ or $\xb1i\xi 1\xi 24$ (as depicted in Fig. 5) by solid circle (red) dots. As *g* approaches the special values *g*_{−1} or *g*_{1}, reaches them, and gets out of the interval $g\u22121,g1$, the corresponding solid circle (red) dots approach the relevant points $\xb1\xi 1\xi 24$ or $\xb1i\xi 1\xi 24$, merge at them, and then split again passing to, respectively, real and imaginary axes, as illustrated by Fig. 5.

### B. Eigenvectors and the Jordan basis

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 $C$:

Namely,

Note that the change of the sign of *G*_{1} of the gyration capacitance or change of the sign of the parameters *b*_{1}, *b*_{2} 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

*Let the circuit be as described in Theorem 3. Then, the sign alteration of the gyration resistance*

*G*

_{1}

*yields a circuit with the evolution matrix*$C\u2212G1$

*that is similar to the evolution matrix*$CG1$

*of the original circuit, that is,*

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:

*Let*C

_{±}

*be*4 × 4

*matrices of the form*

*where*

*ζ*

_{1},

*ζ*

_{2}

*and*

*b*

_{1},

*b*

_{2}

*are complex numbers. Then, matrices*C

_{±}

*can be recast as*

*The Jordan forms*

*J*

_{±}

*of the corresponding matrices*C

_{±}

*are*

*where*$\xb1\zeta 1\zeta 2$

*is one of the values of the square root of*±

*ζ*

_{1}

*ζ*

_{2}

*, and matrices*

*Z*

_{±}

*are*

*Note that the columns of matrices*

*Z*

_{±}

*form the Jordan bases of the corresponding matrices*C

_{±}

*, and the first and the third columns*

*Z*

_{±}

*are the eigenvectors of the corresponding matrices*C

_{±}

*with respective eigenvalues*$\xb1\zeta 1\zeta 2$

*and*$\u2212\xb1\zeta 1\zeta 2$

*.*

*Note also that the matrices*C

_{±}

*and*

*Z*

_{±}

*can be factorized as follows:*

*where matrix*Λ

_{α}

*is the following diagonal matrix:*

In the case when the gyration resistance takes its special values $G\u03071$ 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

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

*Let the circuit be as depicted in*Fig. 2,

*and let all its parameters*

*L*

_{1}

*,*

*C*

_{1}

*,*

*L*

_{2}

*,*

*C*

_{2},

*and*

*G*

_{1}

*be real and non-zero. Then, the companion matrix*$C$

*defined by expression*

*(4.12)*

*has a degenerate eigenvalue if and only if (i) the sign constraints*

*(4.23)*

*are satisfied and (ii) its gyration parameter*

*g*

*takes its two special values,*

*and the corresponding special values of the gyration resistance,*

*where the binary variable*

*σ*

_{1}

*takes values*±1

*. For these special values of the gyration resistance, matrix*$C$

*has exactly two degenerate eigenvalues*±

*s*

_{0}

*of multiplicity*2,

*satisfying the following equations:*

*where*$\xi >0$

*for*

*ξ*> 0

*,*

*δ*= ±1,

*and*

*σ*= ±1

*is the circuit sign index defined by*

*Eq. (4.20)*

*. In addition to that, matrix*$C$

*can be represented as matrix*C

_{±}

*in*

*Eq. (4.47)*

*, with*

*ζ*

_{1},

*ζ*

_{2}

*described by*

*Eq. (4.53)*

*and*

*Consequently, all statements of Lemma 8 for matrix*C

_{±}

*hold including its Jordan form*

*J*

_{±}

*as in*

*Eq. (4.48)*

*and expressions*

*(4.49)*

*for the Jordan basis as columns of matrix*

*Z*

_{±}

*.*

## V. CIRCUIT SYNTHESIS STRATEGY AND ELEMENTS

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:

where $\zeta \u0304$ 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 $\zeta ,\u2212\zeta ,\zeta \u0304,\u2212\zeta \u0304$ consists of (i) two numbers $a,\u2212a$ if *ζ* = *a*, (ii) two numbers $bi,\u2212bi$ if *ζ* = *b*i, and (ii) four numbers

if *ζ* = *a* + *b*i.

To achieve the desired Jordan form for the circuit matrix $H$, we introduce the characteristic polynomial $\chi Hs=\chi Js$ and find its coefficients. Having coefficients *a*_{k} of the polynomial $\chi Hs$ as in Eqs. (2.5) and (2.6), we define the Hamiltonian $Ha$ by the following explicit expression:

Note that the system parameters *a*_{k} can be negative and positive. The particular choice of signs in expression (5.3) is a matter of convenience. The *Hamiltonian*$Ha$*defined by**Eq. (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 $\chi s$ defined by Eq. (2.5) as its characteristic polynomial $\chi Has$, and consequently, the set of the distinct roots

*s*_{j}of the polynomial is exactly the set of all distinct eigenvalues of the circuit matrix $Ha$, that is, $specHa=sj$.Since

*a*_{0}≠ 0, we have*s*_{j}≠ 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*s*_{j}is exactly one, and every*s*_{j}is associated with the single Jordan block $Jnjsj$ of the size*n*_{j}, which is the algebraic multiplicity of eigenvalue*s*_{j}; 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 $\zeta \u2208specHa$ is real or pure imaginary, then the Jordan form $Ja$ of the Hamiltonian matrix $Ha$ has two Jordan blocks $Jn\zeta $ and $Jn\u2212\zeta $ of the matching size

*n*, where*n*is the multiplicity of*ζ*as the root of the polynomial $\chi s$.If $\zeta \u2208specHa$ and

*ζ*=*a*+*b*i, with*a*≠ 0 and*b*≠ 0, then the Jordan form $Ja$ of the Hamiltonian matrix $Ha$ has four Jordan blocks $Jn\xb1a\xb1bi$ and $Jn\xb1a\u2213bi$ of the matching size*n*, where*n*is the multiplicity of*ζ*as the root of the polynomial $\chi s$.

Making particular choices of *a*_{k} 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*:

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

where the Jordan block $Jn\zeta $ is defined by Eq. (1.1).

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

(principal circuit). *Suppose that the principal circuit depicted in* Fig. 1 *has its element values defined by equations (2.7) and (2.8). Then, the dynamics of the principal circuit is governed by the principal Lagrangian* $L$ *defined by* *Eq. (2.2)* *and the principal Hamiltonian* $H$ *defined by defined by* *Eq. (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 of* *n* *involved f-loops and currents* *∂*_{t}*q*_{k}*.*

*The relevant Hamiltonian matrix* $H$ *is cyclic (non-derogatory), and its characteristic polynomial* $\chi s$ *is defined by* *Eq. (2.5)**. The Jordan form* $J$ *of matrix* $H$ *is completely determined by* $\chi s$*. In particular, each distinct root* *s*_{j} *of* $\chi s$ *of the multiplicity* *n*_{j} *is represented in* $J$ *by the single Jordan block* $Jnjsj$ *of the matching size* *n*_{j}*.*

*For particular choices of the monic polynomial* $\chi s$ *as described in Eqs. (5.5) and (5.6), one obtains circuits associated with the Jordan forms represented, respectively, in Eq. (5.7)*.

## VI. THE PRINCIPAL HAMILTONIAN AND LAGRANGIAN

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

where *D*_{a} and *π*_{n} are diagonal *n* × *n* matrices defined by

and *K*_{n} is the *n* × *n* nilpotent matrix defined by

We also make use of the Jordan block $Jn\zeta $ of the size *n* defined by

where $In$ is the *n* × *n* identity matrix.

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

where the system state vector *X* and matrix *M*_{H} are defined by (5.3), and consequently,

With an eigenvalue problem in mind, we introduce the matrix

and then find that the corresponding characteristic function is equal to

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

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 $\chi as$ (see Appendix B), which is the 2*n* × 2*n* matrix defined by

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

where evidently the vector polynomial $Ys$ is uniquely determined by the corresponding eigenvalue *s*. If all eigenvalues *s*_{j}, 1 ≤ *j* ≤ 2*n*, 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 2*n* × 2*n* matrix $Ya$ as the generalized Vandermonde matrix defined by equations (B10) and (B11). Then, according to Proposition 22, we have

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).

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

or equivalently

Note first that $\pi nqs\u22600$; 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:

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

Consequently, we get the following representation for eigenvector $Zs$:

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 *Z*_{k}, 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 $Ys$:

Note that the 2*n* × 2*n* matrix *T*_{a} in Eq. (6.11) is defined by its columns, which are the vector coefficients *Z*_{k} of the vector polynomial $Zs$. Just as the system matrix $Ha$ and the companion matrix $Ca$, matrix *T*_{a} is completely defined by the system parameters *a*_{k} and hence by the polynomial $\chi as$. An analysis shows that *T*_{a} is the 2 × 2 upper triangular block matrix, with blocks of the dimension *n* × *n*, and based on that, one can establish that

The significance of matrix *T*_{a} is that it provides for the similarity relation between the system matrix $Ha$ and its companion matrix $Ca$, that is,

Equations (7.2), (7.3), (7.6), and .(7.7) show examples of matrices $Ha$, $Ca$, and *T*_{a} for the cases *n* = 3, 4.

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

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

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 Lagrangian* $La$ obtained from the principal Hamiltonian $Ha$ by the Legendre transformation is

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

## VII. EXAMPLES OF THE SIGNIFICANT MATRICES FOR THE PRINCIPAL HAMILTONIAN

### A. The principal Hamiltonian and significant matrices for *n* = 3

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

The significant matrices in this case are as follows:

### B. The principal Hamiltonian and significant matrices for *n* = 4

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

The significant matrices in this case are as follows:

## VIII. LAGRANGIAN AND HAMILTONIAN STRUCTURES FOR LINEAR SYSTEMS

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

### A. Lagrangian

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 *∂*_{t}*Q*, that is,

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,

Consequently,

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

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

Note that matrix *θ* enters Eq. (8.5) through its skew-symmetric component $12\theta \u2212\theta T$ justifying as a possibility to impose the skew-symmetry assumption on *θ*, that is,

Indeed, the symmetric part $\theta s=12\theta +\theta T$ of the matrix *θ* is associated with a term to the Lagrangian, which can be recast as is the complete (total) derivative, namely, $12\u2202tQT\theta sQ$. 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 $L\u2192L+\u2202tFq,t$ [Ref. 24 (2.9 and 2.10) and Ref. 25 (I.2)].

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

where

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

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

### B. Hamiltonian

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

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

Consequently,

Note also that Eq. (8.12) imply

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

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

where *M*_{H} is the 2*n* × 2*n* matrix having the block form

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

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

Then, the corresponding to Hamilton vector equation (8.18) matrix similar to the companion polynomial matrix $CAs=sB\u2212A$ in (C5) is

Let us introduce the matrix

implying that the transposed to $H$ matrix $HT$ is similar to $\u2212H$.

### C. Relationship between the Lagrangian and Hamiltonian

implying

and

### D. Lagrangian and Hamiltonian for higher order ODEs

If the Lagrangian $L$ depends on higher order derivatives as in

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

### E. Positive energy case

The main point of this section is that in the case when the energy is non-negative, that is, $HP,Q\u22650$, 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

Note that matrix *M*_{H} can be recast as

where the matrix *K* is the block matrix

which manifestly takes into account the gyroscopic term *θ*. Here, $\alpha $ and $\eta $ 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 *K*_{p}, *K*_{q} are *n* × *n* matrices with real-valued entries with the properties

If we introduce now the force variable

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

where Ω is evidently a self-adjoint operator.

### F. Symplectic and Hamiltonian matrix basics

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

Let $J\u2208R2n\xd72n$ be the unit imaginary matrix defined by Eq. (8.18). It satisfies the identities (8.19).

*symplectic*if it satisfies the following identity (Ref. 29, 3.1):

*T*is nonsingular and

*T*is symplectic if and only if matrices

*T*

^{−1}and

*T*

^{T}are symplectic.

Evidently, symplectic matrices in $R2n\xd72n$ form a group.

*Hamiltonian*(or infinitesimally symplectic) if it satisfies the following identity (Ref. 29, 3.1):

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).

(similarity of a matrix and its transposed). *Let* $A\u2208Cn\xd7n$*. There exists a nonsingular complex symmetric matrix* *S* *such that* *A*^{T} = *SAS*^{−1}*.*

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

(Hamiltonian matrix). *The following are equivalent: (i)* *M* *is Hamiltonian, (ii)* $M=JA$, *where* *A* *is symmetric, and (iii)* $JA$ *is symmetric. Moreover, if* *M* *and* *K* *are Hamiltonian, then so are* *M*^{T}*,* *αM**,* $\alpha \u2208R$*,* *M* ± *K**, and* $M,K\u2261MK\u2212KM$*.*

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

*A matrix*$A\u2208C2n\xd72n$

*is a Hamiltonian matrix if and only if there exist matrices*$A,B,C\u2208Fn\xd7n$

*such that*

*B*

*and*

*C*

*are symmetric and*

*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).

(Similarity of a matrix and its transposed). *Let* $A\u2208Cn\xd7n$*. There exists a nonsingular complex symmetric matrix* *S* *such that* *A*^{T} = *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:

(Jordan structure of a real Hamiltonian matrix). *The characteristic polynomial of a real Hamiltonian matrix is an even polynomial. Thus, if* *ζ* *is an eigenvalue of a Hamiltonian matrix, then* −*ζ**,* $\zeta \u0304$, *and* $\u2212\zeta \u0304$ *are also its eigenvalues with the same multiplicity. The entire Jordan block structure is the same for* *ζ*, −*ζ**,* $\zeta \u0304$, *and* $\u2212\zeta \u0304$*.*

## IX. A SKETCH OF THE BASICS OF ELECTRIC NETWORKS

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.

### A. Circuit elements and their voltage–current relationships

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)]:

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

where *I*_{1}, *I*_{2} and *V*_{1}, *V*_{2} 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.

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:

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:

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:

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

### B. Circuits of negative impedance, capacitance, and inductance

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),

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

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.

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

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

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.

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.

### C. Topological aspects of the electric networks

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***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.**

*4*We denote the number of branches of the network by *N*_{b} ≥ 2 and the number of nodes by *N*_{n} ≥ 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:

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

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

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).

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***, and*

**3***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.*

**5**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 *N*_{n} − 1, and the number KVL equations is *N*_{fl} = *N*_{b} − *N*_{n} + 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

The number *N*_{fl} 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***. There are four links identified by dashed (red) lines and labeled by numbers*

**5****,**

*2***,**

*4***,**

*6***. There are also four oriented f-loops formed by the branches as follows: (1)**

*7**,*

**7***,*

**1***,*

**3***; (2)*

**5***,*

**2***; (3)*

**1***,*

**4***; and (2)*

**3***,*

**6***. 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.*

**5**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).

## X. CONCLUSIONS

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.

## ACKNOWLEDGMENTS

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.

## V. DATA AVAILABILITY

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

### APPENDIX A: JORDAN CANONICAL FORM

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 *n*th order differential equation, we refer to Ref. 42, 25.4.

Let *A* be an *n* × *n* matrix and *λ* be its eigenvalue, and let $r\lambda $ be the least integer *k* such that $NA\u2212\lambda Ik=NA\u2212\lambda Ik+1$, where $NC$ is a null space of a matrix *C*. Then, we refer to $M\lambda =NA\u2212\lambda Ir\lambda $ as the *generalized eigenspace* of matrix *A* corresponding to the eigenvalue *λ*. Then, the following statements hold (Ref. 44, III.4):

*Let*

*A*

*be an*

*n*×

*n*

*matrix and*

*λ*

_{1}, …,

*λ*

_{p}

*be its distinct eigenvalues. Then, generalized eigenspaces*$M\lambda 1,\u2026,M\lambda p$

*are linearly independent, invariant under the matrix*

*A*

*, and*

*Consequently, any vector*

*x*

_{0}

*in*$Cn$

*can be represented uniquely as*

*and*

*where column-vector polynomials*$pjt$

*satisfy*

For a complex number *λ*, a Jordan block $Jr\lambda $ of size *r* ≥ 1 is an *r* × *r* upper triangular matrix of the form

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

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

where *λ*_{j} 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):

*Let*

*A*

*be an*

*n*×

*n*

*matrix. Then, there exists a non-singular*

*n*×

*n*

*matrix*

*Q*

*such that the following block-diagonal representation holds:*

*where*

*J*

*is the Jordan matrix defined by*

*Eq. (A8)*

*and*

*λ*

_{j}

*,*1 ≤

*j*≤

*q*,

*are not necessarily different eigenvalues of matrix*

*A*

*. Representation*

*(A9)*

*is known as the Jordan canonical form of matrix*

*A*

*, and matrices*

*J*

_{j}

*are called Jordan blocks. The columns of the*

*n*×

*n*

*matrix*

*Q*

*constitute the Jordan basis, providing for the Jordan canonical form*

*(A9)*

*of matrix*

*A*

*.*

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

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),

### APPENDIX B: COMPANION MATRIX AND CYCLICITY CONDITION

The companion matrix $Ca$ for the monic polynomial

where coefficients *a*_{k} are complex numbers, is defined by (Ref. 26, 5.2)

Note that

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):

*Let*$A\u2208Cn\xd7n$

*be an*

*n*×

*n*

*matrix with complex-valued entries. Let*$specA=\zeta 1,\zeta 2,\u2026,\zeta r$

*be the set of all distinct eigenvalues, and*$kj=indA\zeta j$

*is the largest size of the Jordan block associated with*

*ζ*

_{j}

*. Then, the minimal polynomial*$\mu As$

*of the matrix*

*A*

*, that is, a monic polynomial of the smallest degree such that*$\mu AA=0$

*, satisfies*

*Furthermore, the following statements are equivalent:*

$\mu As=\chi As=detsI\u2212A$.

*A*is cyclic.For every

*ζ*_{j}, the Jordan form of*A*contains exactly one block associated with*ζ*_{j}.*A*is similar to the companion matrix $C\chi A$.

*Let*$as$

*be a monic polynomial having degree*

*ν*

*and*$Ca$

*is its*

*ν*×

*ν*

*companion matrix. Then, there exist unimodular*

*ν*×

*ν*

*matrices*$S1s$

*and*$S2s$

*, that is,*$detSm=\xb11$

*,*

*m*= 1, 2

*, such that*

*Consequently,*$Ca$

*is cyclic and*

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)].

*Let*$Ca$

*be an*

*n*×

*n*

*a companion matrix of the monic polynomial*$as$

*defined by equation*

*(B1)*

*. Suppose that the set of distinct roots of polynomial*$as$

*is*$\zeta 1,\zeta 2,\u2026,\zeta r$

*and*$n1,n2,\u2026,nr$

*is the corresponding set of the root multiplicities such that*

*Then,*

*where*

*is the Jordan form of the companion matrix*$Ca$

*and the*

*n*×

*n*

*matrix*

*R*

*is the so-called generalized Vandermonde matrix defined by*

*where*

*R*

_{j}

*is an*

*n*×

*n*

_{j}

*matrix of the form*

*As a consequence of representation*

*(B9)*

*,*$Ca$

*is a cyclic matrix.*

As to the structure of matrix *R*_{j} in equation (B11), if we denote by $Y\zeta j$ its first column, then it can be expressed as follows (Ref. 46, 2.11):

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

### APPENDIX C: MATRIX POLYNOMIALS

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

where *s* is complex number, *A*_{k} are constant *m* × *m* matrices, and $x\u2208Cm$ is an *m*-dimensional column-vector. We refer to problem (C1) of funding complex-valued *s* and non-zero vector $x\u2208Cm$ 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 *s* *eigenvector*. Evidently, the characteristic values of problem (C1) can be found from the polynomial *characteristic equation*

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 *s*_{0} as a root of the polynomial $detAs$. In contrast, the *geometric multiplicity* of the eigenvalue *s*_{0} 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,

where *mν* × *mν* matrices A and B are defined by

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 polynomial*$A\lambda $, when $A\nu =I$ is an *m* × *m* identity matrix, matrix B = I is an *mν* × *mν* 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 polynomia*l,” which is an *mν* × *mν* matrix

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 *mν* × *mν* companion polynomial $CAs$ and the original *m* × *m* matrix polynomial $As$, we introduce two *mν* × *mν* matrix polynomials $Es$ and $Fs$. Namely,

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

The matrix polynomial $Fs$ is defined by

Note that both matrix polynomials $Es$ and $Fs$ have constant determinants, readily implying that their inverses $E\u22121s$ and $F\u22121s$ are also matrix polynomials. Then, it is straightforward to verify that

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 $x\u2208Cm\nu $ associated with the companion polynomial $CAs$, that is,

where

Not accidentally, the components of the vector x_{s} in its representation (C12) are in evident relation with the derivatives $\u2202tjx0est=sjx0est$. 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).

*Let*$As$

*as in equations*

*(C1)*

*be regular, that is*, $detAs$

*is not identically zero, and let*

*mν*×

*mν*

*matrices*A

*and*B

*be defined by equations*

*(C2)*

*. Then, the following identities hold:*

*where*$detAs=detsB\u2212A$

*is a polynomial of degree*

*mν*

*if*$detB=detA\nu \u22600$

*. There is one-to-one correspondence between solutions of equations*$Asx=0$

*and*$sB\u2212Ax=0$

*. Namely, a pair*

*s*, x

*solves the eigenvalue problem*$sB\u2212Ax=0$

*if and only if the following equalities hold:*

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 $detB=detA\nu \u22600$, then the degree of the polynomial $detsB\u2212A$ has to be *mν* since A and B are *mν* × *mν* matrices.

Suppose that equations (C15) hold. Then, combining them with the proven identity (C13), we get $sB\u2212Axs=0$, proving that expressions (C15) define an eigenvalue *s* and an eigenvector x = x_{s}.

*x*

_{0}≠ 0. This completes the proof.□

(characteristic polynomial degree)*.* Note that according to Theorem 23, the characteristic polynomial $detAs$ for the *m* × *m* matrix polynomial $As$ has degree *mν*, whereas in the linear case $sI\u2212A0$ for *m* × *m* identity matrix $I$ and *m* × *m* matrix *A*_{0}, the characteristic polynomial $detsI\u2212A0$ 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 *A*_{0}.

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 $A\u2202txt=0$ 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 $sB\u2212Ax=0$, and it has the following immediate implication:

*Under the conditions of Theorem 23 for any eigenvalue*

*s*

_{0}

*, that is,*$detAs0=0$

*, we have*

*In other words, the geometric multiplicities of the eigenvalue*

*s*

_{0}

*associated with matrices*$As0$

*and*

*s*

_{0}B −A

*are equal. In view of identity*

*(C20)*,

*the following inequality holds for the (algebraic) multiplicity*$ms0$:

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.

*Assuming notations introduced in Theorem 23, let us suppose that the multiplicity*$ms0$

*of eigenvalue*

*s*

_{0}

*satisfies*

*Then, the Jordan canonical form of the companion polynomial*$CAs=sB\u2212A$

*has a least one nontrivial Jordan block of the dimension exceeding 2.*

*In particular, if*

*and*$ms0\u22652$,

*then the Jordan canonical form of companion polynomial*$CAs=sB\u2212A$

*has exactly one Jordan block associated with eigenvalue*

*s*

_{0}

*and its dimension is*$ms0$

*.*

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 *s*_{0} is cyclic (nonderogatory) for matrix B^{−1}A, provided B^{−1} exists (see Appendix B).

### APPENDIX D: VECTOR DIFFERENTIAL EQUATIONS AND THE JORDAN CANONICAL FORM

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:

where *A*_{j} are *m* × *m* matrices. Then, differential equation (D1) can be recast in a standard fashion as the *mν*-vector first order differential equation

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

is the *mν*-column-vector function.

In the case when *A*_{ν} is an invertible *m* × *m* matrix, Eq. (D2) can be recast further as

where

Note that one can interpret Eq. (D4) as a particular case of Eq. (D2) where matrices *A*_{ν} and B are identity matrices of the respective dimensions *m* × *m* and *mν* × *mν* and that the polynomial matrix $As$ defined by Eq. (C1) becomes the monic matrix polynomial $A\u0307s$, that is,

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