# Generalized inverse

(Redirected from Pseudo inverse)
"Pseudoinverse" redirects here. For the Moore–Penrose pseudoinverse, sometimes referred to as "the pseudoinverse", see Moore–Penrose pseudoinverse.

In mathematics, a generalized inverse of a matrix A is a matrix that has some properties of the inverse matrix of A but not necessarily all of them. Formally, given a matrix $A \in \mathbb{R}^{n\times m}$ and a matrix $A^{\mathrm g} \in \mathbb{R}^{m\times n}$, $A^{\mathrm g}$ is a generalized inverse of $A$ if it satisfies the condition $AA^{\mathrm g}A = A$.

The purpose of constructing a generalized inverse is to obtain a matrix that can serve as the inverse in some sense for a wider class of matrices than invertible ones. Typically, the generalized inverse exists for an arbitrary matrix, and when a matrix has an inverse, then its inverse and the generalized inverse are the same. Some generalized inverses can be defined in any mathematical structure that involves associative multiplication, that is, in a semigroup.

## Types of generalized inverses

The Penrose conditions are used to define different generalized inverses: for $A \in \mathbb{R}^{n\times m}$ and $A^{\mathrm g} \in \mathbb{R}^{m\times n},$

 1.) $AA^{\mathrm g}A = A$ 2.) $A^{\mathrm g}AA^{\mathrm g}= A^{\mathrm g}$ 3.) $(AA^{\mathrm g})^{\mathrm T} = AA^{\mathrm g}$ 4.) $(A^{\mathrm g}A)^{\mathrm T} = A^{\mathrm g}A$ .

If $A^{\mathrm g}$ satisfies condition (1.), it is a generalized inverse of $A$, if it satisfies conditions (1.) and (2.) then it is a generalized reflexive inverse of $A$, and if it satisfies all 4 conditions, then it is a Moore–Penrose pseudoinverse of $A$.

Other various kinds of generalized inverses include

• One-sided inverse (left inverse or right inverse) If the matrix A has dimensions $n \times m$ and is full rank then use the left inverse if $n > m$ and the right inverse if $n < m$
• Left inverse is given by $A_{\mathrm{left}}^{-1} = \left(A^{\mathrm T} A\right)^{-1} A^{\mathrm T}$, i.e. $A_{\mathrm{left}}^{-1} A = I_m$ where $I_m$ is the $m \times m$ identity matrix.
• Right inverse is given by $A_{\mathrm{right}}^{-1} = A^{\mathrm T} \left(A A^{\mathrm T}\right)^{-1}$, i.e. $A A_{\mathrm{right}}^{-1} = I_n$ where $I_n$ is the $n \times n$ identity matrix.
• Drazin inverse
• Bott–Duffin inverse
• Moore–Penrose pseudoinverse

## Uses

Any generalized inverse can be used to determine if a system of linear equations has any solutions, and if so to give all of them.[1] If any solutions exist for the n × m linear system

$Ax=b$

with vector x of unknowns and vector b of constants, all solutions are given by

$x=A^{\mathrm g}b + [I-A^{\mathrm g}A]w$

parametric on the arbitrary vector w, where $A^{\mathrm g}$ is any generalized inverse of $A.$ Solutions exist if and only if $A^{\mathrm g}b$ is a solution – that is, if and only if $AA^{\mathrm g}b=b.$