Page 103 - Textos de Matemática Vol. 44

ON THE SPECTRUM OF POLYNOMIALS OF TWO NON-COMMUTATIVE MATRICES
NAHUM KRUPNIK AND YAFIM SPIGEL
Dedicated to Professor Nata´lia Bebiano’s 60th anniversary with the best wishes
Abstract. In this paper we describe some classes of polynomials F(A,B), such that the spectrum of F (A, B) coincides with the spectrum of F (B, A) for all pairs of matrices A, B with entries from C.
1. Introduction
Denote by N the set of natural numbers, by C the set of complex numbers and by Mn the algebra of all n × n matrices with entries from C.
Consider the polynomial F(A,B) := AB + γE , where A,B ∈ Mn, E is a unit matrix, and γ ∈ C. It is well known (see, for example, equality (2.3) below), that
(1.1) spec F (A, B) = spec F (B, A),
where spec(A) := {λ1, λ2, ..., λn} is the set of all eigenvalues of the matrix A with multiplicities taken into account. It is an interesting problem to describe all polynomials F(A,B), for which equality (1.1) holds. This paper is devoted to the mentioned problem.
Definition 1.1. For the purpose of this article we say that a polynomial F(A,B) is n−admissible if the equality (1.1) holds for any pair of matrices A, B ∈ Mn . We say that a polynomial F is admissible if it is n−admissible for all n ∈ N. Also in this paper we say that a polynomial F(A,B) is trivial, if F(A,B) = F(B,A).
The following two theorems give a complete description of all admissible polynomials of degrees two and three.
2010 Mathematics Subject Classification. Primary 11C20; Secondary 11C08
Key words and phrases. trace, spectrum
The research of the first author was partially supported by Retalon Inc., Toronto, ON, Canada.
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