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

Abstract

The Hadamard and Kronecker products of two n×m matrices A,B are related by A∘B=P T 1(A⊗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∘B)(A∘B) *⩽AA *∘BB * is extended to (A∘B)(A∘B) *⩽ 1 2 (AA *∘BB *+AB *∘BA *)⩽AA *∘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∘B, ∥A∘B∥⩽ 1 2 (∥A∥ 2∥B∥ 2+∥AB *∥ 2) 1/2⩽∥A∥ ∥B∥ and an improvement on the weak majorization of A∘B, σ 2(A∘B)≺ w 1 2 σ 2(A)· σ 2(B)+ σ 2(AB) ≺ w σ 2(A)∘ σ 2(B). 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: κ(DXE)⩾(2∥X∘X −T∥ 2−∥X∘X −T∥ 2) 1/2⩾∥X∘X T−1∥. For A⩾0, we have (I∘A) 2⩽ 1 2 (I∘A 2+A∘A)⩽I∘A 2 and (A 1/2∘A −1/2) 2⩽ 1 2 (I+A∘A −1)⩽A∘A −1 when A>0. The latter inequality is compared to Styan's inequality (A∘A) −1⩽ 1 2 (I+A∘A −1) when A is a correlation matrix and is shown to possess stronger properties of ordering. Finally, the relationship A∘B=P T 1(A⊗B)P 2 is applied to determine conditions of singularity of certain orderings of the Hadamard products of matrices.