generalized inverse

Let $A$ 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

AA^{-}A=A.

Examples

1.

Let

$A=\begin{pmatrix}2&3&0\\ 1&2&0\\ 0&0&0\end{pmatrix}.$

Then any matrix of the form

$A^{-}=\begin{pmatrix}2&-3&a\\ -1&2&b\\ c&d&e\end{pmatrix},$

where $a, b, c, d$ and $e\in\mathbb{C}$ , is a generalized inverse.
2.

Using the same example from above, if $a=b=c=d=e=0$ , then we have an example of the Moore-Penrose generalized inverse, which is a unique matrix.
3.

Again, using the example from above, if $a=b=c=d=0$ and $e$ 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

\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 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

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

Title	generalized inverse
Canonical name	GeneralizedInverse
Date of creation	2013-03-22 14:31:26
Last modified on	2013-03-22 14:31:26
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	5
Author	CWoo (3771)
Entry type	Definition
Classification	msc 15A09
Classification	msc 62J10
Classification	msc 62J12