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 ${\displaystyle AA^{\mathrm {g} }A=A.}$^[1][2][3] A generalized inverse exists for an arbitrary matrix, and when a matrix has a regular inverse, this inverse is its unique generalized inverse.^[1]

Motivation

Consider the linear system

${\displaystyle Ax=y}$

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

${\displaystyle AA^{-1}A=A.}$

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

${\displaystyle AGy=y.}$^[4]

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

${\displaystyle AGA=A.}$

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 ${\displaystyle AGA=A.}$‍^[1][2][3] The matrix ${\displaystyle A^{-1}}$ has been termed a regular inverse of ${\displaystyle A}$ by some authors.^[5]

Types

Important types of generalized inverse include:

• One-sided inverse (right inverse or left inverse)
  • Right inverse: If the matrix ${\displaystyle A}$ has dimensions ${\displaystyle n\times m}$ and ${\displaystyle {\textrm {rank}}(A)=n}$, then there exists an ${\displaystyle m\times n}$ matrix ${\displaystyle A_{\mathrm {R} }^{-1}}$ called the right inverse of ${\displaystyle A}$ such that ${\displaystyle AA_{\mathrm {R} }^{-1}=I_{n}}$, where ${\displaystyle I_{n}}$ is the ${\displaystyle n\times n}$ identity matrix.
  • Left inverse: If the matrix ${\displaystyle A}$ has dimensions ${\displaystyle n\times m}$ and ${\displaystyle {\textrm {rank}}(A)=m}$, then there exists an ${\displaystyle m\times n}$ matrix ${\displaystyle A_{\mathrm {L} }^{-1}}$ called the left inverse of ${\displaystyle A}$ such that ${\displaystyle A_{\mathrm {L} }^{-1}A=I_{m}}$, where ${\displaystyle I_{m}}$ is the ${\displaystyle m\times m}$ identity matrix.^[6]
• Bott–Duffin inverse
• Drazin inverse
• Moore–Penrose inverse

Some generalized inverses are defined and classified based on the Penrose conditions:

1. ${\displaystyle AA^{\mathrm {g} }A=A}$
2. ${\displaystyle A^{\mathrm{g}}}$Zdroj:https://en.wikipedia.org?pojem=Generalized_inverse
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