On the invertibility of matrices with a double saddle-point structure

TL;DR

This paper provides necessary and sufficient conditions for the invertibility of symmetric block matrices with a double saddle-point structure, including explicit formulas for their inverses under various rank conditions.

Contribution

It introduces a comprehensive set of criteria for invertibility and explicit inverse formulas for double saddle-point matrices, extending previous results to rank-deficient cases.

Findings

Derived necessary and sufficient invertibility conditions.

Provided explicit inverse formulas under various rank conditions.

Extended analysis to cases with rank-deficient diagonal blocks.

Abstract

We establish necessary and sufficient conditions for invertibility of symmetric three-by-three block matrices having a double saddle-point structure \fb{that guarantee the unique solvability of double saddle-point systems}. We consider various scenarios, including the case where all diagonal blocks are allowed to be rank deficient. Under certain conditions related to the nullity of the blocks and intersections of their kernels, an explicit formula for the inverse is derived.