Reducibility theorems for pairs of matrices as rational criteria

Abstract

Theorems giving conditions for a pair of matrices to be reducible to a special form by a simultaneous similarity transformation such as the classical McCoy's theorem or theorems due to Shapiro and Watters are traditionally perceived as nonconstructive ones. We disprove this perception by showing that conditions of each of the theorems above can be verified using a finite number of arithmetic operations. A new extension of McCoy's theorem is stated which, in some respects, is more convenient than Shapiro's theorem.