Simultaneous similarity of matrices

Abstract

In this paper we solve completely and explicitly the long-standing problem of classifying pairs of n × n complex matrices ( A, B) under the simultaneous similarity ( TAT −1, TBT −1). Roughly speaking, the classification decomposes to a finite number of steps. In each step we consider an open algebraic set M 0 n,2, r,π ⊆ M n × M n ( M n = the set of n × n complex-valued matrices). Here r and π are two positive integers. Then we construct a finite number of rational functions ø 1,…,ø s in the entries of A and B whose values are constant on all pairs similar in M n,2, r,π to ( A, B). The values of the functions ø i ( A, B), i = 1,…, s, determine a finite number (at most κ( n, 2, r)) of similarity classes in M n,2, r,π . Let S n be the subspace of complex symmetric matrices in M n . For ( A, B) ϵ S n × S n we consider the similarity class ( TAT t, TBT t ), where T ranges over all complex orthogonal matrices. Then the characteristic polynomial | λI − ( A + xB)| determines a finite number of similarity classes for almost all pairs ( A, B) ϵ S n × S n .