A quantitative version of the observation that the Hadamard product is a principal submatrix of the Kronecker product
It is assumed that singular values \(\sigma_\nu(A)\) of a matrix \(A\) are decreasingly ordered. The notation \(\sigma(A)= (\sigma_1(A), \dots, \sigma_n(A))^T\) is used. Let \({\mathbf x}\) and \({\mathbf y}\) be real vectors whose entries are such that \(x_1\geq x_2\geq \cdots\geq x_n\) and \(y_1\geq y_2\geq \cdots \geq y_n\). Then \({\mathbf x}\) is said to weakly majorize \({\mathbf y}\) if \(\sum^k_{i =1} y_i\leq\sum^k_{i=1}x_i\) for \(k=1,\dots,n\), and it is written by \({\mathbf y} <_w {\mathbf x}\). The Hadamard and Kronecker products of two \(n\times m\) matrices \(A,B\) are related by \(A\circ B=P^T_1\) \((A\otimes B) P_2\), 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. The result \[ (A \circ B)(A\circ B)^*\leq AA^* \circ BB^* \] is extended to \[ (A\circ B)(A\circ B)^* \leq{1\over 2} (AA^* \circ BB^*+ AB^*\circ BA^*) \leq AA^*\circ BB^*. \] This leads to \[ \|A \circ B\|\leq\Bigl( \textstyle {1\over 2} \|A\|^2 \|B\|^2 +\|AB^* \|^2 \Bigr)^{1/2}\leq \|A\|\|B\| \] and an improvement on the weak majorization of \(A\circ B\), \[ \sigma^2(A\circ B)<_w \textstyle {1\over 2} \bigl(\sigma^2 (A)\cdot \sigma^2(B) +\sigma^2 (AB) \bigr)<_w \sigma^2(A) \circ \sigma^2(B). \] For a real nonsingular matrix \(X\) and invertible diagonal matrices \(D,E\) the spectral condition number \(\kappa(\cdot)\) is shown to be \[ \kappa (DXE)\geq \bigl(2 \|X\circ X^{-T}\|^2 -\|X\circ X^{-T}\|^2 \bigr)^{1/2}\geq \|X\circ X^{-T} \|. \] Finally, the relationship \(A\circ B=P^T_1\) \((A\otimes B)P_2\) is applied to determine conditions of singularity of certain orderings of the Hadamard products of matrices.
scientific article; zbMATH DE number 1420031
scientific article