The exponential of an N × N matrix can always be expressed as a matrix polynomial of order N − 1. In particular, a general group element for the fundamental representation of SU(N) can be expressed as a matrix polynomial of order N − 1 in a traceless N × N hermitian generating matrix, with polynomial coefficients consisting of elementary trigonometric functions dependent on N − 2 invariants in addition to the group parameter. These invariants are just angles determined by the direction of a real N-vector whose components are the eigenvalues of the hermitian matrix. Equivalently, the eigenvalues are given by projecting the vertices of an N 1 -simplex onto a particular axis passing through the center of the simplex. The orientation of the simplex relative to this axis determines the angular invariants and hence the real eigenvalues of the matrix.

1.
T. L.
Curtright
and
C. K.
Zachos
, “
Elementary results for the fundamental representation of SU(3)
,”
Rep. Math.
(unpublished); e-print arXiv:1508.00868 [math.RT]. Earlier versions were first available from Research Gate.
2.
M.
Gell-Mann
and
Y.
Ne’eman
,
The Eightfold Way
(
W. A. Benjamin
,
1964
), also see https://en.wikipedia.org/wiki/Gell-Mann_matrices.
3.
Y.
Lehrer
, “
On functions of matrices
,”
Rend. Circolo Mat. Palermo
6
,
103
108
(
1957
);
Y.
Lehrer–Ilamed
, “
On the direct calculations of the representations of the three-dimensional pure rotation group
,”
Math. Proc. Cambridge Philos. Soc.
60
,
61
66
(
1964
) (especially see Remark (1) and Eqn(10)).
4.
A. J.
MacFarlane
,
A.
Sudbery
, and
P. H.
Weisz
, “
On Gell-Mann’s λ-matrices, d- and f-tensors, octets, and parametrizations of SU(3)
,”
Commun. Math. Phys.
11
,
77
90
(
1968
).
5.
S. P.
Rosen
, “
Finite transformations in various representations of SU(3)
,”
J. Math. Phys.
12
,
673
681
(
1971
).
6.
D.
Kusnezov
, “
Exact matrix expansions for group elements of SU(N)
,”
J. Math. Phys.
36
,
898
906
(
1995
).
7.
A.
Laufer
, “
The exponential map of GL(N)
,”
J. Phys. A: Math. Gen.
30
,
5455
(
1997
).
8.
H. S. M.
Coxeter
,
Regular Polytopes
, 3rd ed. (
Dover
,
2012
).
9.
See https://en.wikipedia.org/wiki/Cubic_function#Three_real_rootsand references therein, especially,
R. W. D.
Nickalls
, “
Viète, Descartes and the cubic equation
,”
Math. Gaz.
90
,
203
208
(
2006
).
10.
A.
Cayley
, “
A memoir on the theory of matrices
,”
Philo. Trans. Roy. Soc. Lon.
CXLVIII
,
17
37
(
1858
), although the paper is reprinted in his collected works as indicated in the original version of the reference. It seems that Cayley himself edited that volume of his collected works.
11.
The general method to express anyanalytic matrix function of a finite, diagonalizable matrix as a polynomial in the matrix, through the use of projection matrices, is due to,
J. J.
Sylvester
,
Philos. Mag.
16
,
267
269
(
1883
), also see https://en.wikipedia.org/wiki/Sylvester’s_formula.
12.
This is a well-known fact. For example, see Section 3.1 in
T. L.
Curtright
and
D. B.
Fairlie
, “
A Galileon primer
,” e-print arXiv:1212.6972 [hep-th].
13.
M. X.
He
and
P. E.
Ricci
, “
On Taylor’s formula for the resolvent of a complex matrix
,”
Comput. Math. Appl.
56
,
2285
2288
(
2008
).
14.
T. S.
Van Kortryk
, “
Cayley transforms of s u 2 representations
,” e-print arXiv:1506.00500 [math-ph].
15.
T. L.
Curtright
, “
More on rotations as spin matrix polynomials
,”
J. Math. Phys.
56
,
091703
(
2015
); e-print arXiv:1506.04648 [math-ph].
16.

The same procedure may be used to construct sequentially a hierarchy of polynomials for the exponentials of almost any matrix, as in (1).

17.
V. Y.
Pan
, “
Solving a polynomial equation: Some history and recent progress
,”
SIAM Rev.
39
,
187
220
(
1997
).
18.
T. L.
Curtright
,
D. B.
Fairlie
, and
C. K.
Zachos
, “
A compact formula for rotations as spin matrix polynomials
,”
SIGMA
10
,
084
(
2014
), e-print arXiv:1402.3541 [math-ph].
19.
T. L.
Curtright
and
T. S.
Van Kortryk
, “
On rotations as spin matrix polynomials
,”
J. Phys. A: Math. Theor.
48
,
025202
(
2015
); e-print arXiv:1408.0767 [math-ph].
20.

Although the eigenvalues themselves still define a circle with the same radius as before, namely r2, since k=13cos2θ+2πk/3=3/2.

21.
B. G.
Wybourne
,
Classical Groups for Physicists
(
John Wiley & Sons
,
1974
).
22.

Here is the solution to the exercise: The tetrahedron’s vertices are at a distance 3r/2 from the origin, such that λ1=12r1,1,1ê, λ2=12r1,1,1ê, λ3=12r1,1,1ê, and λ4=12r1,1,1ê, where êcosθ,sinθcosϕ+π/4,sinθsinϕ+π/4.

23.
See https://en.wikipedia.org/wiki/Quartic_function and references therein, especially,
R. W. D.
Nickalls
, “
The quartic equation: Invariants and Euler’s solution revealed
,”
Math. Gaz.
93
,
66
75
(
2009
).
24.
R. B.
King
,
Beyond the Quartic Equation
(
Birkhäuser
,
1996
);
B.
Sturmfels
, “
Solving algebraic equations in terms ofA-hypergeometric series
,”
Discrete Math.
210
,
171
181
(
2000
);
O.
Nash
, “
On Klein’s icosahedral solution of the quintic
,”
Expo. Math.
32
,
99
120
(
2014
).
You do not currently have access to this content.