Generalized inverses

In mathematics, and in particular, algebra, a generalized inverse of an element x is an element y that has some properties of an inverse element but not necessarily all of them. 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 ${\displaystyle A}$.

Formally, given a matrix ${\displaystyle A\in \mathbb {R} ^{n\times m}}$ and a matrix ${\displaystyle A^{\mathrm {g} }\in \mathbb {R} ^{m\times n}}$, ${\displaystyle A^{\mathrm {g} }}$ is a generalized inverse of ${\displaystyle A}$ if it satisfies the condition ${\displaystyle AA^{\mathrm {g} }A=A}$.

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