A quantitative version of the observation that the Hadamard product is a principal submatrix of the Kronecker product

The Hadamard and Kronecker products of two n x m matrices A, B are related by A circle B = P-1(T) (A x B) P-2, where P-1, P-2 are partial permutation matrices. After establishing several properties of the P matrices, this relationship is employed to demonstrate how a simplified theory of the Hadamard product can be developed. During this process the well-known result (A circle B)(A circle B)* less than or equal to AA* circle BB* is extended to (A circle B)(A circle B)* less than or equal to 1/2(AA* circle BB* + AB* circle BA*) less than or equal to AA* circle BB* showing an inherent link between the Hadamard product and conventional product of two matrices. This leads to a sharper bound on the spectral norm of A circle B, \\A circle B\\ less than or equal to (1/2(\\A\\(2)\\B\\(2) + \\AB*\\(2)))(1/2) less than or equal to \\A\\ \\B\\ and an improvement on the weak majorization of A circle B, <(sigma)under bar>(2)(A circle B) <(omega) 1/2 (<(sigma)under bar>(2)(A) . <(sigma)under bar>(2)(B) + <(sigma)under bar>(2)(AB)) <(omega) <(sigma)under bar>(2) (A) circle <(sigma)under bar>(2)(B). For a real non-singular matrix X and invertible diagonal matrices D, E the spectral condition number kappa(.) is shown to be, if scaled, bounded below as follows: kappa(DXE) greater than or equal to (2\\X circle X-T\\(2) - \\X circle X-T\\(1/2))(1/2) greater than or equal to \\X circle X-T\\. For A greater than or equal to 0, we have (I circle A)(2) less than or equal to 1/2 (I circle A(2) + A circle A) less than or equal to I circle A(2) and (A(1/2) circle A(-1/2))(2) less than or equal to 1/2(I + A circle A(-1)) less than or equal to A circle A(-1) when A > 0. The latter inequality is compared to Styan's inequality (A circle A)(-1) less than or equal to 1/2(I + A circle A(-1)) when A is a correlation matrix and is shown to possess stronger properties of ordering. Finally, the relationship A circle B = P-1(T)(A x B)P-2 is applied to determine conditions of singularity of certain orderings of the Hadamard products of matrices. (C) 2000 Elsevier Science Inc. All rights reserved.