PlanetMath: Moore-Penrose generalized inverse

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Moore-Penrose generalized inverse

(Definition)

Let be an $m\times n$ matrix with entries in $\mathbb{C}$ . The Moore-Penrose generalized inverse, denoted by $A^{\dagger}$ , is an $n\times m$ matrix with entries in $\mathbb{C}$ , such that

$AA^{\dagger}A=A$
$A^{\dagger}AA^{\dagger}=A^{\dagger}$
$AA^{\dagger}$ and $A^{\dagger}A$ are both Hermitian

Remarks

The Moore-Penrose generalized inverse of a given matrix is unique.
If $A^{\dagger}$ is the Moore-Penrose generalized inverse of , then $(A^{\dagger})^{\operatorname{T}}$ is the Moore-Penrose generalized inverse of $A^{\operatorname{T}}$ .
If such that
1. $A\in\mathbb{C}^{m\times n}$ , $B\in\mathbb{C}^{m\times r}$ , and $C\in\mathbb{C}^{r\times n}$ ,
2. $r=\operatorname{rank}(A)=\operatorname{rank}(B)=\operatorname{rank}(C)$ , then
  $\displaystyle A^{\dagger}=C^{\ast}(CC^{\ast})^{-1}(B^{\ast}B)^{-1}B^{\ast}.$

For example, let

$\displaystyle A=\begin{pmatrix}1&1&i \\ 0&1&0 \end{pmatrix}.$

Transform

to its row echelon form to get a decomposition of

, where

$\displaystyle B=\begin{pmatrix}1&1 \\ 0&1 \end{pmatrix}$ and $\displaystyle C=\begin{pmatrix}1&0&i \\ 0&1&0 \end{pmatrix}.$

It is readily verified that $2=\operatorname{rank}(A)=\operatorname{rank}(B)=\operatorname{rank}(C)$ . So

$\displaystyle A^{\dagger}=\frac{1}{2}\begin{pmatrix}1&-1 \\ 0&2 \\ -i&i \end{pmatrix}.$

We check that

$\displaystyle AA^{\dagger}=I$ and $\displaystyle A^{\dagger}A=\frac{1}{2}\begin{pmatrix}1&0&i \\ 0&2&0 \\ -i&0&1 \end{pmatrix}$

are both Hermitian. Furthermore, $AA^{\dagger}A=A$ and $A^{\dagger}AA^{\dagger}=A^{\dagger}$ . So, $A^{\dagger}$ is the Moore-Penrose generalized inverse of

"Moore-Penrose generalized inverse" is owned by CWoo.

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See Also: Drazin inverse, pseudoinverse

Other names:

Moore-Penrose pseudoinverse

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Cross-references: decomposition, row echelon form, Transform, Hermitian, matrix
There is 1 reference to this entry.

This is version 5 of Moore-Penrose generalized inverse, born on 2004-08-03, modified 2006-11-03.
Object id is 6067, canonical name is MoorePenroseGeneralizedInverse.
Accessed 11997 times total.

Classification:

AMS MSC:	15A09 (Linear and multilinear algebra; matrix theory :: Matrix inversion, generalized inverses)
	60J10 (Probability theory and stochastic processes :: Markov processes :: Markov chains with discrete parameter)

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