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Inverting the sum of two singular matrices

Sofia Eriksson, Jonas Nordqvist

TL;DR

This work addresses inverting a sum of a singular matrix and a structured low-rank perturbation, $\widetilde{\mathbf{A}}=\mathbf{A}+\mathbf{e}D\mathbf{f}^*$, under the conditions that $D$ and $\widetilde{\mathbf{A}}$ are invertible and $\operatorname{rank}(\widetilde{\mathbf{A}})=\operatorname{rank}(\mathbf{A})+\operatorname{rank}(\mathbf{e}D\mathbf{f}^*)$. The main contribution is an explicit inverse of $\widetilde{\mathbf{A}}$ in the form $\widetilde{\mathbf{A}}^{-1}=\mathbf{G}+\mathbf{x}D^{-1}\mathbf{y}^*$, with $\mathbf{G}$, $\mathbf{x}$ and $\mathbf{y}$ computed in a way that does not depend on $D$; the result is first established via singular value decomposition and then recast into a non-SVD formulation. The work clarifies connections to generalized inverses, showing partial satisfaction of Penrose conditions and relating $\mathbf{G}$ to the Moore–Penrose inverse $\mathbf{A}^+$ through projections, while also linking to prior results (e.g., Riedel, 1992). Additionally, a singular-matrix determinant lemma is derived, giving $\det(\widetilde{\mathbf{A}})=\det(\mathbf{A}+\mathbf{e}\mathbf{f}^*)\det(D)$ and $\det(\widetilde{\mathbf{A}}^{-1})=\det(\mathbf{G}+\mathbf{x}\mathbf{y}^*)\det(D^{-1})$, thereby extending Woodbury-type identities to singular bases and aiding practical inversion in applications such as SBP-SAT finite difference schemes.

Abstract

Square matrices of the form $\widetilde{\mathbf{A}} =\mathbf{A} + \mathbf{e}D \mathbf{f}^*$ are considered. An explicit expression for the inverse is given, provided $\widetilde{\mathbf{A}}$ and $D$ are invertible with $\text{rank}(\widetilde{\mathbf{A}}) =\text{rank}(\mathbf{A})+\text{rank}(\mathbf{e}D \mathbf{f}^*)$. The inverse is presented in two ways, one that uses singular value decomposition and another that depends directly on the components $\mathbf{A}$, $\mathbf{e}$, $\mathbf{f}$ and $D$. Additionally, a matrix determinant lemma for singular matrices follows from the derivations.

Inverting the sum of two singular matrices

TL;DR

This work addresses inverting a sum of a singular matrix and a structured low-rank perturbation, , under the conditions that and are invertible and . The main contribution is an explicit inverse of in the form , with , and computed in a way that does not depend on ; the result is first established via singular value decomposition and then recast into a non-SVD formulation. The work clarifies connections to generalized inverses, showing partial satisfaction of Penrose conditions and relating to the Moore–Penrose inverse through projections, while also linking to prior results (e.g., Riedel, 1992). Additionally, a singular-matrix determinant lemma is derived, giving and , thereby extending Woodbury-type identities to singular bases and aiding practical inversion in applications such as SBP-SAT finite difference schemes.

Abstract

Square matrices of the form are considered. An explicit expression for the inverse is given, provided and are invertible with . The inverse is presented in two ways, one that uses singular value decomposition and another that depends directly on the components , , and . Additionally, a matrix determinant lemma for singular matrices follows from the derivations.
Paper Structure (5 sections, 3 theorems, 29 equations)

This paper contains 5 sections, 3 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. The inverse of the sum of two singular matrices
  3. Relations to generalized inverses
  4. Expressing the inverse without singular value decomposition
  5. Singular matrix determinant lemma

Key Result

Theorem 2.1

Let $n,k$ be integers such that $n>k\geq1$. Consider the complex matrix $\widetilde{\mathbf{A}}=\mathbf{A}+\mathbf{e}D \mathbf{f}^*,$ where $\mathbf{A}, \mathbf{e}$, $D$ and $\mathbf{f}$ are $n\times n$, $n\times k$, $k\times k$ and $n\times k$ respectively. Further, suppose the rank of $\mathbf{A}$ with $\mathbf{G}$, $\mathbf{x}$ and $\mathbf{y}$ given in MWZsvdparts.

Theorems & Definitions (6)

  • Theorem 2.1
  • proof : Proof of Theorem \ref{['thmsvd']}
  • Corollary 1: Corollary of Theorem \ref{['thmsvd']}
  • proof : Proof of Corollary \ref{['cor1']}
  • Remark 1
  • Lemma 1