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

The Hadamard and Kronecker products of two n×m matrices A,B are related by Aring operatorB=PT1(Acircle times operatorB)P2, where P1,P2 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 (Aring operatorB)(Aring operatorB)*less-than-or-equals, slantAA*ring operatorBB* is extended to image 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 Aring operatorB, image and an improvement on the weak majorization of Aring operatorB, image For a real non-singular matrix X and invertible diagonal matrices D,E the spectral condition number κ(·) is shown to be, if scaled, bounded below as follows: image κ(DXE)greater-or-equal, slanted(2short parallelXring operatorX−Tshort parallel2−short parallelXring operatorX−Tshort parallel2)1/2greater-or-equal, slantedshort parallelXring operatorXT−1short parallel. For Agreater-or-equal, slanted0, we have image and image when A>0. The latter inequality is compared to Styanʹs inequality image when A is a correlation matrix and is shown to possess stronger properties of ordering. Finally, the relationship Aring operatorB=PT1(Acircle times operatorB)P2 is applied to determine conditions of singularity of certain orderings of the Hadamard products of matrices.