generalized inverse
Let $A$ be an $m\times n$ matrix with entries in $\u2102$. A generalized inverse, denoted by ${A}^{}$, is an $n\times m$ matrix with entries in $\u2102$, such that
$$A{A}^{}A=A.$$ 
Examples

1.
Let
$$A=\left(\begin{array}{ccc}\hfill 2\hfill & \hfill 3\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 2\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right).$$ Then any matrix of the form
$${A}^{}=\left(\begin{array}{ccc}\hfill 2\hfill & \hfill 3\hfill & \hfill a\hfill \\ \hfill 1\hfill & \hfill 2\hfill & \hfill b\hfill \\ \hfill c\hfill & \hfill d\hfill & \hfill e\hfill \end{array}\right),$$ where $a,b,c,d$ and $e\in \u2102$, 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 MoorePenrose 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^{}
$${\text{\mathbf{X}}}^{\mathrm{T}}\text{\mathbf{X}}\bm{\beta}={\text{\mathbf{X}}}^{\mathrm{T}}\text{\mathbf{Y}},$$ 
to get the MLE $\widehat{\bm{\beta}}$ of the parameter vector $\bm{\beta}$. 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 ${\text{\mathbf{X}}}^{\mathrm{T}}\text{\mathbf{X}}$ is singular. Then the MLE can be given by
$$\widehat{\bm{\beta}}={({\text{\mathbf{X}}}^{\mathrm{T}}\text{\mathbf{X}})}^{}{\text{\mathbf{X}}}^{\mathrm{T}}\text{\mathbf{Y}}.$$ 
Title  generalized inverse 

Canonical name  GeneralizedInverse 
Date of creation  20130322 14:31:26 
Last modified on  20130322 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 