Generalized Schur complements involving the Kronecker products of positive semidefinite matrices (Q2191942)

Schur complements have been extensively studied and their origin can be traced back to Issai Schur. The name was coined by Emilie Virginia Haynsworth for a square nonsingular matrix. An important reference which is a survey in the area is [\textit{F. Zhang} (ed.), The Schur complement and its applications. New York, NY: Springer (2005; Zbl 1075.15002)], which is not listed in the reference. For \(A\in {\mathbb C}^{m\times n}\), \(\ell\in \min \{m, n\}\) and \(\alpha, \beta \subset L =\{1, 2, \dots, \ell\}\), the generalized Schur complement of the principal submatrix \(A(\alpha, \alpha)\) is \[ A/\alpha = A(\alpha',\beta') -A(\alpha', \alpha)[A(\alpha)]^+A(\alpha, \beta'), \] where \(A(\alpha, \beta)\) denotes the submatrix of \(A\) contained in the rows indexed by \(\alpha\) and the columns indexed by \(\beta\), \(\alpha'\) denotes the complement of \(\alpha\) in \(L\), and \(A^+\) denotes the Moore-Penrose inverse of \(A\). In this paper, some matrix equalities and inequalities for generalized Schur complements of Kronecker product of two positive semidefinite matrices are obtained. Let \(A\in {\mathbb C}^{m\times m}\), \(B\in {\mathbb C}^{n\times n}\) positive semidefinite matrices and \(\alpha\subset \{1, \dots, m\}\), \(\beta \subset \{1, \dots, n\}\), and \(X = \{1, 2, \dots, mn\}\). Then the following are the main results in Section 3. \begin{itemize} \item[1.] (Theorem 1) \((A\otimes B)/\gamma = (A/\alpha)\otimes (B/\beta),\) where \(\gamma = X- \gamma'\), \(\gamma' = \{ n(i-1)+ j : i\in \alpha', j\in \beta'\}\). \item[2.] (Theorems 2--4) \((A/\alpha)\otimes (B/\beta) \ge (A\otimes B)/\delta\), where \(\delta = X-\delta'\), \(\delta' = \{ n(i-1)+ j : i\in \alpha, j\in \beta'\}\), \(\delta' = \{ n(i-1)+ j : i\in \alpha', j\in \beta\}\), or \(\delta' = \{ n(i-1)+ j : i\in \alpha, j\in \beta\}\) and \(\le\) is the Loewner order. \end{itemize} The above results appeared in the author's paper [J. Yunnan Univ., Nat. Sci. 39, No. 5, 719--726 (2017; Zbl 1399.15034)], which is not listed in the reference. Section 4 contains some inequalities for eigenvalues of generalized Schur complements of Kronecker product of two positive definite matrices. They are derived from interlacing inequalities and the results in Section 3.