All our matrices are square with real elements. The Schur product of two n × n matrices B = (bij) and C = (cij) (i, j, = 1, 2, …, n), is an n × n matrix A = (aij) with aij = bij cij, (i, j = 1, 2, …, n).
A result due to Schur [1] states that if B and C are symmetric positive definite matrices then so is their Schur product A. A question now a rises. Can any symmetric positive definite matrix be expressed as a Schur product of two symmetric positive definite matrices? The answer is in the affirmative as we show in the following theorem.