A note on pairs of matrices with product zero

Abstract

The independence of YI and Y2 is, in a straightforward way, equivalent toA+B havingeigenvalues Al ... , An, and Theorem I (which was first noted by Craig [3]) is sufficiently fundamental that generally it is now at least stated in advanced texts. For example, a portion of a proof is given in [4]. Apparently in ignorance of [3,4,5], an alternate proof of Theorem I is given in [1]. Our goal is to give a generalization of Theorem I whose proof is quite simple. In addition to including a rat]~er different proof of Theorem I, our observation points out that the symmetry of A and B is not an essential assumption. We recall that the singular values of a general complex matrix A are, by definition, the nonnegative square roots of the eigenvalues of A*A. A good general reference on the singular values decomposition of a matrix is [6].