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 ${X}^{\operatorname{T}}{X}$ is singular. Then the MLE can be given by $$\hat{\boldsymbol{\beta}}=(\textbf{X}^{\operatorname{T}}\textbf{X})^{-}\textbf{X}^{\operatorname{T}}\textbf{Y}.$$