Some Simple Criteria for the Solvability of Block $2 \times 2$ Linear Systems

Abstract

In many areas, there arise linear systems of the form

$$\begin{aligned} \left[ \begin{array}{ll} A &{}\quad B^\top \\ B &{}\quad -\,D \\ \end{array} \right] \left[ \begin{array}{c} x \\ y \\ \end{array} \right] =\left[ \begin{array}{c} f \\ g \\ \end{array} \right] , \end{aligned}$$

where $A \in {\mathbf {R}}^{n\times n}, D \in {\mathbf {R}}^{p\times p}$ are symmetric and positive semi-definite and $B \in {\mathbf {R}}^{p \times n}.$ In this paper, some simple criteria for this special linear systems to have solutions and the unique solution are provided, and the solvability conditions are expressed by $A, B, D, f$ and $g$.