Schur's theorem for a block Hadamard product
Let \(M_{m,n}\) be the space of complex \(m\times n\) matrices, and let \(\mathbf{M}_{p,q}(M_{m,n})\) be the space of \(p\times q\) block matrices \(\mathbf{A}=(A_{\alpha\beta})_{\alpha=1,\dots,p}^{\beta=1,\dots,q}\), \(A_{\alpha\beta}\in M_{m,n}\). The authors define a block Kronecker product of \(A\in M_{m,l}\) and \(\mathbf{B}=(B_{\mu\nu})\in\mathbf{M}_{s,t}(M_{l,n})\) by \(A\,\boxtimes\,\mathbf{B}=(AB_{\mu\nu})_{\mu=1,\dots,s}^{\nu=1,\dots,t}\). Further, if \(\mathbf{A}=(A_{\alpha\beta})\in\mathbf{M}_{p,q}(M_{m,l})\), they define \(\mathbf{A}\,\boxtimes\,\mathbf{B}=(A_{\alpha\beta}\boxtimes\mathbf{B})_{\alpha=1,\dots,p}^{\beta=1,\dots,q}\). Let \(\mathbf{A}\) be as before and \(\mathbf{B}=(B_{\alpha\beta})\in\mathbf{M}_{p,q}(M_{l,n})\). \textit{R.~A.~Horn}, \textit{R.~Mathias} and \textit{Y.~Nakamura} [Linear Multilinear Algebra 30, No. 4, 303--314 (1991; Zbl 0754.15015)] defined their block Hadamard product by \(\mathbf{A}\,\square\,\mathbf{B}=(A_{\alpha\beta}B_{\alpha\beta})_{\alpha=1,\dots,p}^{\beta=1,\dots,q}\). The present authors study~\(\boxtimes\) and apply it in studying~\(\square\). Under strong commutation assumptions, they generalize Schur's product theorem and Oppenheim's determinantal inequality.
scientific article; zbMATH DE number 6044835
scientific article