An application of the Cayley-Hamilton theorem to matrix polynomials in several variables

Abstract

Let Σ={m 1}|Σ|2 be a finite set of square n × n matrices. Let Lp be the linear space with rank rp (n 2) spanned by the set of matrices (m:meσ∗,|m| p and let c(σ) be defined by c(σ) = min {p:rp =r p+1, where |X| denotes the number of elements in a set σ or the length of a word σ and σ denotes the set of all finite words over σ (here a word is a sequence of matrices from σ and is understood as the matrix product of the matrices in the sequence). It is shown in the paper that C(σ) L(n 2+2)/31. The proof of the above bound is based on combinatorial arguments and it is not known whether the bound is sharp. It follows from the Cayley-Hamiiton Theorem that c(σ)n−1if σ a singleton. It is also shown (a result of H. W. Lenstra Jr.) that the bound C(σ)n−1 holds for commutative matrices.