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

Fatemeh P. A. Beik, Chen Greif, Manfred Trummer

TL;DR

This work analyzes invertibility and solvability of symmetric three-by-three block matrices with a double saddle-point form $\mathcal{K}$ built from blocks $A,D,E$ and off-diagonal blocks involving $B$ and $C$. It establishes necessary and sufficient invertibility conditions under potentially rank-deficient diagonal blocks, and provides explicit inverse formulas in key scenarios, notably when ${\rm null}(A)=m$ and $E$ is nonsingular. By linking invertibility to kernel intersections and Schur-like reductions, the paper extends known results and offers constructive representations useful for preconditioning and solver design in applications such as PDE discretizations and electromagnetics. The findings enable rigorous solvability analysis and pave the way for structure-exploiting preconditioners that can accelerate Krylov subspace methods for double saddle-point systems. The work thus contributes both theoretical insights and practical tools for computational science problems governed by such block-structured systems.

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.

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

TL;DR

This work analyzes invertibility and solvability of symmetric three-by-three block matrices with a double saddle-point form built from blocks and off-diagonal blocks involving and . It establishes necessary and sufficient invertibility conditions under potentially rank-deficient diagonal blocks, and provides explicit inverse formulas in key scenarios, notably when and is nonsingular. By linking invertibility to kernel intersections and Schur-like reductions, the paper extends known results and offers constructive representations useful for preconditioning and solver design in applications such as PDE discretizations and electromagnetics. The findings enable rigorous solvability analysis and pave the way for structure-exploiting preconditioners that can accelerate Krylov subspace methods for double saddle-point systems. The work thus contributes both theoretical insights and practical tools for computational science problems governed by such block-structured systems.

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.
Paper Structure (6 sections, 15 theorems, 67 equations)

This paper contains 6 sections, 15 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Necessary and sufficient conditions on invertibilty
  3. Invertibility when the $(1,1)$-block is maximally rank deficient
  4. On the nullity of the $(3,3)$ diagonal block
  5. An explicit formula for the inverse
  6. Concluding remarks and future work

Key Result

Proposition 2.1

BG2022 The following conditions are necessary for $\mathcal{K}$ to be invertible: In the case where $A$ is nonsingular, sufficient conditions for $\mathcal{K}$ to be invertible are that $S_1=D+BA^{-1}B^T$ and $S_2=E+C\textcolor{black}{S}_1^{-1}C^T$ are invertible.

Theorems & Definitions (25)

  • Proposition 2.1
  • Theorem 2.1
  • proof
  • Corollary 2.1
  • proof
  • Corollary 2.2
  • Corollary 2.3
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • ...and 15 more