Let F be a Held and F[x] the ring of polynomials in an indeterminate .\ over F. Let, J.l l1(F). Al l1 (F[x]) denote the algebras of n x II matrices over F. F[x]. respectively. and GL(u, F). GL{.r:, F[x]) their corresponding groups of units. Given A(x). B(x) E .~11/(F[xJ). we say that A(s). H(x) arc PSβ’equivulcnt (:= "polynomial-scalar") if there exist P(x) E GL (Il\ F[x])~Q E GL(Il, F) with B(x) :::: P(x)A(x)Q. We consider the problem of determining whether A(x) and B(x) are PS-equivalcnt. In other words we wish to classify the orbits of M,,(F{xJ) under the action of GL(n. FIx]) x GL(Il, F) acting via
We observe that the classical problems of determining the simultaneous equivalence of two k-tupJes of elements of M,,(F) and the simultaneous similarity of two k-tupIcs of clements of M,,(F) arc special cases of this problem. We observe that the Smith invariants of A(x) and B(x) (that is, invariants for t he action of
Base-' on this we present a ncar canonical form for PS-cquivatencc and an algorithm for (; ":: " ' I , '1, . ~whether two matrices in ncar canonical form arc PSβ’equivalent. We exa: . . . . ' "' " ' " j the "generic case" in which A(x) has a single Smith invariant different fror. . . . "' I ," ',in a further set of invariants in this case, and based on these we present an imrroved aigorithm to determine PSβ’cquivalcncc in this situation. While the