is the product of two nonnegative matrices. It follows from Corollary 2.3 that the same is true for T,B,. Since B, is positive, we conclude that T, is the product of three nonnegative matrices. n 4. FOUR NONNEGATIVE MATRICES THEOREM 4.1. Any singular square matrix is the product of four non- negative matrices, and four is the smallest such number. Moreover, any three of these matrices may be taken to be positive. Proof. Let T be an n x n singular matrix. By the Jordan canonical form, T is similar to a matrix T’ of the form R@S, where