Generalized 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. A generalized inverse exists for an arbitrary matrix, and when a matrix has an inverse, then this inverse is its unique generalized inverse. 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. 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 $AA^{\mathrm g}b=b.$

See also

• Inverse element

External links

• 15A09 Matrix inversion, generalized inverses in Mathematics Subject Classification, MathSciNet search