PlanetMath: generalized inverse

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generalized inverse

(Definition)

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

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

Examples

Let
$\displaystyle A=\begin{pmatrix}2&3&0 \\ 1&2&0 \\ 0&0&0 \end{pmatrix}.$
Then any matrix of the form
$\displaystyle A^{-}=\begin{pmatrix}2&-3&a \\ -1&2&b \\ c&d&e \end{pmatrix},$
where and $e\in\mathbb{C}$ , is a generalized inverse.
Using the same example from above, if , then we have an example of the Moore-Penrose generalized inverse, which is a unique matrix.
Again, using the example from above, if and is any complex number, we have an example of a Drazin inverse.

Remark Generalized inverse of a matrix has found many applications in statistics. For example, in general linear model, one solves the set of normal equations

$\displaystyle \textbf{X}^{\operatorname{T}}\textbf{X}\boldsymbol{\beta}=\textbf{X}^{\operatorname{T}}\textbf{Y},$

to get the MLE $\hat{\boldsymbol{\beta}}$ of the parameter vector $\boldsymbol{\beta}$ . If the design matrix $\textbf{X}$ is not of full rank (this occurs often when the model is either an ANOVA or ANCOVA type) and hence $\textbf{X}^{\operatorname{T}}\textbf{X}$ is singular. Then the MLE can be given by

$\displaystyle \hat{\boldsymbol{\beta}}=(\textbf{X}^{\operatorname{T}}\textbf{X})^{-}\textbf{X}^{\operatorname{T}}\textbf{Y}.$

"generalized inverse" is owned by CWoo.

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Cross-references: singular, type, ANCOVA, ANOVA, rank, design, vector, parameter, MLE, normal equations, general linear model, statistics, applications, Drazin inverse, complex number, Moore-Penrose generalized inverse, matrix
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This is version 2 of generalized inverse, born on 2004-08-03, modified 2004-08-03.
Object id is 6065, canonical name is GeneralizedInverse.
Accessed 8024 times total.

Classification:

AMS MSC:	15A09 (Linear and multilinear algebra; matrix theory :: Matrix inversion, generalized inverses)
	62J10 (Statistics :: Linear inference, regression :: Analysis of variance and covariance)
	62J12 (Statistics :: Linear inference, regression :: Generalized linear models)

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