# Generalized inverse

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 $X \in \mathbb{R}^{m\times n}$, $X$ is a generalized inverse of $A$ if it satisfies the condition $A X A = A$.

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 $X \in \mathbb{R}^{m\times n}$,

 1.) $AXA = A$ 2.) $XAX = X$ 3.) $(AX)^{T} = AX$ 4.) $(XA)^{T} = XA$ .

If $X$ 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 pseudoinverse of $A$.

Other various kinds of generalized inverses include

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