In mathematics, and in particular, algebra, a generalized inverse (or, g-inverse) of an element x is an element y that has some properties of an inverse element but not necessarily all of them. The purpose of constructing a generalized inverse of a matrix is to obtain a matrix that can serve as an inverse in some sense for a wider class of matrices than invertible matrices. Generalized inverses can be defined in any mathematical structure that involves associative multiplication, that is, in a semigroup. This article describes generalized inverses of a matrix ${\displaystyle A}$.

A matrix ${\displaystyle A^{\mathrm {g} }\in \mathbb {R} ^{n\times m}}$ is a generalized inverse of a matrix ${\displaystyle A\in \mathbb {R} ^{m\times n}}$ if ${\displaystyle AA^{\mathrm {g} }A=A.}$ A generalized inverse exists for an arbitrary matrix, and when a matrix has a regular inverse, this inverse is its unique generalized inverse.

Consider the linear system

where ${\displaystyle A}$ is an ${\displaystyle m\times n}$ matrix and ${\displaystyle y\in {\mathcal {C}}(A),}$ the column space of ${\displaystyle A}$. If ${\displaystyle m=n}$ and ${\displaystyle A}$ is nonsingular then ${\displaystyle x=A^{-1}y}$ will be the solution of the system. Note that, if ${\displaystyle A}$ is nonsingular, then

Now suppose ${\displaystyle A}$ is rectangular (${\displaystyle m\neq n}$), or square and singular. Then we need a right candidate ${\displaystyle G}$ of order ${\displaystyle n\times m}$ such that for all ${\displaystyle y\in {\mathcal {C}}(A),}$

That is, ${\displaystyle x=Gy}$ is a solution of the linear system ${\displaystyle Ax=y}$. Equivalently, we need a matrix ${\displaystyle G}$ of order ${\displaystyle n\times m}$ such that

Hence we can define the generalized inverse as follows: Given an ${\displaystyle m\times n}$ matrix ${\displaystyle A}$, an ${\displaystyle n\times m}$ matrix ${\displaystyle G}$ is said to be a generalized inverse of ${\displaystyle A}$ if ${\displaystyle AGA=A.}$‍ The matrix ${\displaystyle A^{-1}}$ has been termed a regular inverse of ${\displaystyle A}$ by some authors.

Important types of generalized inverse include: