On the similarity classes among product S of m nonsingular matrices in various orders

It is shown that for any k is an element of {1,2,..., (m - 1)!} there exist m invertible complex matrices such that among the m! products A(sigma) = A(sigma(1))A(sigma(2)) . . . A(sigma(m)) , sigma is an element of S-m, exactly k different similarity classes occur. The cases in which the matrices A(i), are upper triangular or are 2-by-2 are considered in detail. In the former case, it is shown that any m - 1 unispectral matrices with common eigenvalue may be found among the A(sigma), and in the latter case it is shown explicitly how to achieve (m - 1)! and (m - 1)!/2 similarity classes, as well as any number from 1 to 6 when m = 4. Other particular results are given, as well as a discussion of further natural questions. (C) 2014 Elsevier Inc. All rights reserved.