A property of equivalence

Abstract

Let R be a prin cipal id eal rin g. W e write A E B, if A and B a re matrices over R wh ich are equivale nt (see [1] for a co mple te di scussion of thi s topic). The Kronecke r product of any two matrices A and B will be de noted by A @ B. The follow in g res ult was s ugges ted by a re mark made by W. D. Wallis in hi s s urvey paper [2]: THEOREM: Suppose that K, A, Bare nonsinguLar matrices over R such that K @ A E K @ B. Then AEB. It is not actu ally necessary to assume that A and Bare nonsin gular , bu t doin g so simpljfies the exposi ti on. We firs t prove th e followin g: LEMMA: Suppose that the sets