Simultaneous similarity of matrices

Abstract : In this paper we solve completely and explicitly the long standing problem of classifying pairs of nxn complex matrices (A,B) under a simultaneous similarity. Roughly speaking, the classification decomposes to a finite number of steps. In each step we consider an open algebraic set. Then we construct a finite number of rational functions in the entries of A and B whose values are constant on all pairs similar to (A,B). The values of the functions phi sub i (A,B), i equals 1,...,s, determine a finite number of similarity classes.