一类具有正定Schur补的非埃尔米特矩阵
On a class of non-Hermitian matrices with positive definite Schur complements
摘要：
给定正定矩阵$A\in \mathbb{C}^{n\times n}$和埃尔米特矩阵$D\in \mathbb{C}^{m\times m}$，本文刻划了在什么条件下存在严格压缩矩阵$K\in \mathbb{C}^{n\times m}$，使得非Hermite块矩阵\[ \left[ \begin{array}{cc} A & -AK \\ K^*A & D \end{array} \right] \]关于其子矩阵$A$有正定Schur补.此外，我们表明，~ $K$可以选择这样的块矩阵的对角化是有保证的，我们计算其频谱。此外，我们显示了一个连接到最近开发的框架理论克莱因空间。
英文摘要：
Given a positive definite matrix $A\in \mathbb{C}^{n\times n}$ and a Hermitian matrix $D\in \mathbb{C}^{m\times m}$, we characterize under which conditions there exists a strictly contractive matrix $K\in \mathbb{C}^{n\times m}$ such that the non-Hermitian block-matrix \[ \left[ \begin{array}{cc} A & -AK \\ K^*A & D \end{array} \right] \] has a positive definite Schur complement with respect to its submatrix~$A$. Additionally, we show that~$K$ can be chosen such that diagonalizability of the block-matrix is guaranteed and we compute its spectrum. Moreover, we show a connection to the recently developed frame theory for Krein spaces.
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