generalized inverse

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

$$A{A}^{-}A=A.$$

Examples

1. Let

   Then any matrix of the form

   where $a,b,c,d$ and $e\in ℂ$, 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.

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

$${\text{𝐗}}^{\mathrm{T}}\text{𝐗}𝜷={\text{𝐗}}^{\mathrm{T}}\text{𝐘},$$

to get the MLE $\widehat{𝜷}$ of the parameter vector $𝜷$. 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 ${\text{𝐗}}^{\mathrm{T}}\text{𝐗}$ is singular. Then the MLE can be given by

$$\widehat{𝜷}={({\text{𝐗}}^{\mathrm{T}}\text{𝐗})}^{-}{\text{𝐗}}^{\mathrm{T}}\text{𝐘}.$$