Appendix P — The Woodbury Matrix Identity – Bayesian Statistics

Keywords

Woodbury matrix identity, linear equations, matrix factorization, numerical stability, linear algebra

P.1 The Woodbury Matrix Identity

The Woodbury matrix identity is a method for efficiently updating the inverse of a matrix when it is modified by a low-rank update. It is particularly useful in numerical linear algebra and statistics.

Definition P.1 (Definition of the Woodbury Matrix Identity) The Woodbury matrix identity, also known as the matrix inversion lemma, provides a formula for computing the inverse of a matrix that has been modified by a low-rank update. Specifically, if A is an invertible n \times n matrix, U is an n \times k matrix, V is a k \times n matrix, and C is an invertible k \times k matrix, the Woodbury matrix identity states that: (A + UCV)^{-1} = A^{-1} - A^{-1}U(C^{-1} + VA^{-1}U)^{-1}VA^{-1}.

This identity allows us to compute the inverse of the modified matrix A + UCV efficiently, especially when k is much smaller than n, as it avoids the need to compute the inverse of the larger matrix directly.

Some properties of the Woodbury matrix identity include:

It is particularly useful for solving linear equations of the form (A + UCV)x = b when A is large and U, V, and C represent a low-rank update.
It can be used to derive efficient algorithms for matrix inversion and linear regression, especially in cases where the data matrix has a low-rank structure.
It is closely related to the Sherman-Morrison formula, which is a special case of the Woodbury matrix identity when k = 1.
It can be used to improve the numerical stability of matrix computations by avoiding the direct inversion of large matrices, which can be computationally expensive and prone to numerical errors.

---
title: "The Woodbury Matrix Identity"
subtitle: "Appendix"
description: "This appendix explains the Woodbury matrix identity, which is a method for efficiently updating the inverse of a matrix when it is modified by a low-rank update."
categories: 
    - "numerical linear algebra"
keywords: 
    - "Woodbury matrix identity"
    - "linear equations"
    - "matrix factorization"
    - "numerical stability"
    - "linear algebra"
---


## The Woodbury Matrix Identity {#sec-woodbury}

The Woodbury matrix identity is a method for efficiently updating the inverse of a matrix when it is modified by a low-rank update. It is particularly useful in numerical linear algebra and statistics.

::: {#def-woodbury-identity}
## Definition of the Woodbury Matrix Identity

The Woodbury matrix identity, also known as the matrix inversion lemma, provides a formula for computing the inverse of a matrix that has been modified by a low-rank update. Specifically, if $A$ is an invertible $n \times n$ matrix, $U$ is an $n \times k$ matrix, $V$ is a $k \times n$ matrix, and $C$ is an invertible $k \times k$ matrix, the Woodbury matrix identity states that:
$$
(A + UCV)^{-1} = A^{-1} - A^{-1}U(C^{-1} + VA^{-1}U)^{-1}VA^{-1}.
$$

This identity allows us to compute the inverse of the modified matrix $A + UCV$ efficiently, especially when $k$ is much smaller than $n$, as it avoids the need to compute the inverse of the larger matrix directly.

Some properties of the Woodbury matrix identity include:

a. It is particularly useful for solving linear equations of the form $(A + UCV)x = b$ when $A$ is large and $U$, $V$, and $C$ represent a low-rank update.
b. It can be used to derive efficient algorithms for matrix inversion and linear regression, especially in cases where the data matrix has a low-rank structure.
c. It is closely related to the Sherman-Morrison formula, which is a special case of the Woodbury matrix identity when $k = 1$.
d. It can be used to improve the numerical stability of matrix computations by avoiding the direct inversion of large matrices, which can be computationally expensive and prone to numerical errors.   



:::