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generalized inverse
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(Definition)
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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
- 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.
- 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.
- 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 $$\hat{\boldsymbol{\beta}}=(\textbf{X}^{\operatorname{T}}\textbf{X})^{-}\textbf{X}^{\operatorname{T}}\textbf{Y}.$$
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"generalized inverse" is owned by CWoo.
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Cross-references: singular, type, ANCOVA, ANOVA, rank, design, vector, parameter, MLE, normal equations, general linear model, statistics, applications, Drazin inverse, complex number, Moore-Penrose generalized inverse, matrix
There are 2 references to this entry.
This is version 2 of generalized inverse, born on 2004-08-03, modified 2004-08-03.
Object id is 6065, canonical name is GeneralizedInverse.
Accessed 9491 times total.
Classification:
| AMS MSC: | 15A09 (Linear and multilinear algebra; matrix theory :: Matrix inversion, generalized inverses) | | | 62J10 (Statistics :: Linear inference, regression :: Analysis of variance and covariance) | | | 62J12 (Statistics :: Linear inference, regression :: Generalized linear models) |
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Pending Errata and Addenda
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