Kronecker Square Roots and the Block Vec Matrix

Using the block vec matrix, I give a necessary and sufficient condition for factorization of a matrix into the Kronecker product of two other matrices. As a consequence, I obtain an elementary algorithmic procedure to decide whether a matrix has a square root for the Kronecker product. Introduction My statistician colleague, J.E. Chacón, asked me how to decide if a real given matrix A has a square root for the Kronecker product (i.e., if there exists a B such that A = B ⊗ B) and, in the positive case, how to compute it. His questions were motivated by the fact that, provided that a certain real positive definite symmetric matrix has a Kronecker square root, explicit asymptotic expressions for certain estimator errors could be obtained. See [1], for a discussion of the importance of multivariate kernel density derivative estimation. This note is written mostly due to the lack of a suitable reference for the existence of square roots for the Kronecker product, and it is organized as follows: first of all, I study the problem of the factorization of a matrix into a Kronecker product of two matrices, by giving a necessary and sufficient condition under which this happens (Theorem 3). As a preparation for the main result, I introduce the block vec matrix (Definition 1). Now, the block vec matrix and Theorem 3 solve our problem in a constructive way. 1. Kronecker product factorization Throughout this note N, R, and C denote the sets of non-negative integers, real numbers, and complex numbers, respectively. All matrices considered here have real or complex entries; A denotes the transpose of A and tr(A) denotes its trace. The operator that transforms a matrix into a stacked vector is known as the vec operator (see, [3, Definition 4.2.9] or [6, § 7.5]). If A = (