Order invariant spectral properties for several matrices
Given a set \(\left\{ A_{1},\dots,A_{m}\right\} \) of \(n\times n\) matrices over a field \(F\), consider the set consisting of the \(m!\) products of the matrices in every possible order. If the matrices are nonsingular then any two products related by a cyclic permutation are similar, but in general these \((m-1)!\) similarity classes may all have different traces. When the matrices are singular the situation is more complicated, even for \(m=2\) [see \textit{H. Flanders}, Proc. Am. Math. Soc. 2, 871--874 (1951; Zbl 0044.00602)]. In the present paper the authors consider three situations: (SOI) all products are in the same similarity class; (EOI) all products have the same characteristic polynomial; and (TOI) all products have the same trace. The authors prove various properties of these classes, mainly for \(m=3\). A sample theorem is the following. If \(m=3\), \(n=2\) and \(char(F)\neq2\), then (TOI) holds if and only if \(A_{1},A_{2}\) and \(A_{3}\) are simultaneously symmetrizable or simultaneously upper triangularizable. The analogous result is false when \(m>3\).
scientific article; zbMATH DE number 5677359
scientific article