Matrix inverses. Sometimes we need them. Worse, sometimes we need generalizations of them, inverses of singular (a.k.a. non-invertible) matrices. This page is for notes on that theme.
But first, please fill out this form.
✅ By reading further I acknowledge that a matrix inverse is actually what I want, even though the more common case would be that I would like the solution of a linear system without forming the inverse.
OK, done? Let us continue.
1 All possible generalizations of matrix inverses
Many (Rao and Mitra 1973; Ben-Israel and Greville 2003). Nick Higham’s introduction generalized inverses is an easier start.
A beautiful, compact summary is in Searle (2014):
…there are (with one exception) many matrices
satisfying which is condition (i). Each matrix satisfying AGA = A is called a generalized inverse of A, and if it also satisfies it is a reflexive generalized inverse. The exception is when is nonsingular: there is then only one ; namely, . 8.2 Arbitrariness That there are many matrices can be illustrated by showing ways in which from one others can be obtained. Thus, if A is partitioned as where is nonsingular with the same rank as , then is a generalized inverse of for any values of and . This can be used to show that a generalized inverse of a symmetric matrix is not necessarily symmetric; and that of a singular matrix is not necessarily singular. A simpler illustration of arbitrariness is that if
is a generalized inverse of then so is for any values of and .
2 Moore-Penrose pseudo-inverse
A classic. The “default” generalized inverse.
See Nick Higham’s What Is the Pseudoinverse of a Matrix?
Let us do the conventional thing and mention which properties of the pseudo-inverse are shared by the inverse:
The pseudo-inverse (or Moore-Penrose inverse) of a matrix
symmetric symmetric
Want more? As always, we copy-paste the omnibus results of Petersen and Pedersen (2012):
Assume
I find definition in terms of these properties totally confusing. Perhaps a better way of thinking about pseudo-inverses is via their action upon vectors. TBC
Consider the Moore-Penrose of
Meta note: In general, proving things about pseudo-inverses by the constructive solution given by the SVD is much more compact than via the algebraic properties, as well as more intuitive, at least for me.
There is a cute special-case result for low rank matrices.
3 (Generalized) Bott-Duffin pseudo-inverse
Many variants (Wu et al. 2023; Gao et al. 2023; Liu, Wang, and Wei 2009).
4 Drazin inverse
The Drazin inverse is introduced in a mediocre Wikipedia article:
Let
be square matrix. The index of is the least nonnegative integer such that . The Drazin inverse of is the unique matrix that satisfies It’s not a generalized inverse in the classical sense, since in general.
- If
is invertible with inverse , then . - If
is a block diagonal matrix
where is invertible with inverse and is a nilpotent matrix, then
- Drazin inversion is invariant under conjugation. If
is the Drazin inverse of , then is the Drazin inverse of . - The Drazin inverse of a matrix of index 0 or 1 is called the group inverse or
-inverse and denoted . The group inverse can be defined, equivalently, by the properties , and . - A projection matrix
, defined as a matrix such that , has index 1 (or 0) and has Drazin inverse . - If
is a nilpotent matrix (for example a shift matrix), then .
5 Incrementally updating
6 Low-rank
See low-rank matrices.