Quasi-normal matrices and products

Abstract

A normal matrix A = (aij) with complex elements is a matrix such that AACT = ACTA where ACT denotes the (complex) conjugate transpose of A. In an article by K. Morita [2] a quasi-normal matrix is defined to be a complex matrix A which is such that AACT = ATAC, where T denotes the transpose of A and AC the matrix in which each element is replaced by its conjugate, and certain basic properties of such a matrix are developed there. (Some doubt might exist concerning the use of ‘quasi’ since this class of matrices does not contain normal matrices as a sub-class; however, in deference to the original paper and the normal canonical form of Theorem 1 below, the terminology in [2] is used.)