Page 105 - Textos de Matemática Vol. 44

SPECTRUM OF POLYNOMIALS OF TWO MATRICES 95
Below by Ch(X) we denote the characteristic polynomial of the matrix X, by tr X the trace of this matrix, by E the unit matrix and by Eik the “matrix units”, i.e., Eik = [ars] , where aik = 1 and ars = 0 otherwise.
2. Some sufficient conditions
In this section we assume that F (A, B) is a polynomial of any degree n ≥ 2 and propose some sufficient conditions for F(A,B) to be admissible. We start with the
Theorem 2.1. Let F(A,B) admit a following representation:
(2.1) F (A, B) = F1(A, B)F2(A, B) + dE,
where F1(A, B), F2(A, B) are some polynomials, d is a complex number and
(2.2) F1(A, B)F2(A, B) = F2(B, A)F1(B, A)
Then the polynomial F(A,B) is admissible.
Proof. It is known (see, for example, [2, p. 9]) that for any two matrices X, Y ∈ Mn and any complex number c the following equality holds:
(2.3) det(XY − cE) = det(YX − cE)
For completeness we present here the proof of this equality.
For c = 0 equality (2.3) is trivial; for c ̸= 0 it follows from the equality
E 0 E 0 E aY E aY E−aYX 0 E 0
(2.4) X E 0 E−aXY 0 E = 0 E 0 E X E , where a = 1/c.
Now let a polynomial F (A, B) admit representation (2.1) with condition (2.2). For an arbitrary λ ∈ C we denote c = λ − d. Using successively relations (2.1), (2.3), (2.2), we obtain
det (F (A, B) − λE) = det (F1(A, B)F2(A, B) − cE) = det (F2(A, B)F1(A, B)) − cE) = det (F1(B, A)F2(B, A)) − cE)
= det (F (B, A) − λE) ,
i.e., specF(A,B) = specF(B,A).