On sums and products of Matrix similarity classes I*

Abstract

We consider the problem under which conditions the product or sum of similarity classes of two given matrices contains all cyclic matrices. This problem was already considered in (C. Kurtz (2001). On the products of diagonal conjugacy classes. Communications in Algebra, 29(2), 769–779) for products of similarity classes of diagonal matrices, but now we are capable of proving a much stronger result even for diagonal matrices but with a method which also applies to nondiagonal matrices, and to sums of similarity classes. Our proof proceeds by explicitly constructing a square matrix H(λ) of size n, where is a vector of variables, such that all but n entries are fixed elements of the ground field; the last n entries are identified with the variables . The matrix H(λ) will then have the property that the set (respectively ) contains at least one matrix of every similarity class of cyclic matrices.