On simultaneous similarity of matrices and related questions Original Research Article

Let F be a field and F[x] the ring of polynomials in an indeterminate x over F. Let Mn(F), Mn(F[x]) denote the algebras of n × n matrices over F, F[x], respectively, and GL(n,F), GL(n,F[x]) their corresponding groups of units. Given A(x), B(x) set membership, variant Mn(F[x]), we say that A(x), B{x) are PS-equivalent ( = “polynomial-scalar”) if there exist P(x) set membership, variant GL(n,F[x]), Q set membership, variant GL(n,F) with B(x) = P(x)A(x)Q. We consider the problem of determining whether A(x) and B(x) are PS-equivalent. In other words we wish to classify the orbits of Mn(F[x]) under the action of GL(n,F[x]) × GL(n,F) acting via (T(x),Q)A(x) = T(x)−1A(x)Q. We observe that the classical problems of determining the simultaneous equivalence of two k-tuples of elements of Mn(F) and the simultaneous similarity of two k-tuples of elements of Mn(F) are special cases of this problem. We observe that the Smith invariants of A(x) and B(x) (that is, invariants for the action of GL(n,F[x]) × GL(n,F[x]) on Mn(F[x]) via (T(x),S(x))A(x) = T(x)−1A(x)S(x)) must be equal if A(x), B(x) are PS-equivalent. Based on this we present a near canonical form for PS-equivalence and an algorithm for determining whether two matrices in near canonical form are PS-equivalent. We examine in detail the “generic case” in which A(x) has a single Smith invariant different from 1 and obtain a further set of invariants in this case, and based on these we present an improved algorithm, to determine PS-equivalence in this situation. While the main emphasis in the paper is on finding a reasonably good algorithm in the generic ease, we also discuss the question of finding a complete set of invariants for PS-equivalence, especially in the case n = 2. where connections with linear fractional transformations arise. A much more comprehensive account of the invariants in the simultaneous similarity problem can be found in Friedlandʹs paper (Adv. in Math. 50 (1983) 189–265).