generalized inverse


Let A be an m×n matrix with entries in . A generalized inverse, denoted by A-, is an n×m matrix with entries in , such that

AA-A=A.

Examples

  1. 1.

    Let

    A=(230120000).

    Then any matrix of the form

    A-=(2-3a-12bcde),

    where a,b,c,d and e, 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 statisticsMathworldMathworldPlanetmath. For example, in general linear model, one solves the set of normal equationsMathworldPlanetmath

𝐗T𝐗𝜷=𝐗T𝐘,

to get the MLE 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 𝐗T𝐗 is singular. Then the MLE can be given by

𝜷^=(𝐗T𝐗)-𝐗T𝐘.
Title generalized inverse
Canonical name GeneralizedInverse
Date of creation 2013-03-22 14:31:26
Last modified on 2013-03-22 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