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



  1. 1.



    Then any matrix of the form


    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


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

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