On a Criterion for Simultaneous Block-Diagonalization of Normal Matrices

Abstract

Assume that A 1 , … , A s are complex normal n × n matrices, p is a natural number and S 2p is the standard polynomial in 2p non-commutative variables. It follows from classical results of S. Amitsur, J. Levitzki and H. Shapiro that A 1 , … , A s can be simultaneously block-diagonalized by a unitary matrix with blocks of sizes not greater than p if and only if the algebra generated by A1, … , A s satisfies the polynomial identity S 2 p = 0. We call this theorem the ALS-criterion for simultaneous block-diagonalization of normal matrices. In this paper, we present some application of the ALS-criterion in quantum theory. Namely, we give another proof of the renowned Morris-Shore transformation. Moreover, we discuss computable versions of the ALS-criterion. These versions allow one to verify the condition S 2 p = 0 in a finite number of steps. Such an approach is more useful in practical applications than the original one.