# 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. 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. 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. 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 GeneralizedInverse 2013-03-22 14:31:26 2013-03-22 14:31:26 CWoo (3771) CWoo (3771) 5 CWoo (3771) Definition msc 15A09 msc 62J10 msc 62J12