Moore-Penrose generalized inverse

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

  1. 1.


  2. 2.


  3. 3.

    AA and AA are both Hermitian


  • The Moore-Penrose generalized inverse of a given matrix is unique.

  • If A is the Moore-Penrose generalized inverse of A, then (A)T is the Moore-Penrose generalized inverse of AT.

  • If A=BC such that

    1. (a)

      Am×n, Bm×r, and Cr×n,

    2. (b)

      r=rank(A)=rank(B)=rank(C), then


For example, let


Transform A to its row echelon formMathworldPlanetmath to get a decomposition of A=BC, where

B=(1101) and C=(10i010).

It is readily verified that 2=rank(A)=rank(B)=rank(C). So


We check that

AA=I and AA=12(10i020-i01)

are both Hermitian. Furthermore, AAA=A and AAA=A. So, A is the Moore-Penrose generalized inverse of A.

Title Moore-Penrose generalized inverse
Canonical name MoorePenroseGeneralizedInverse
Date of creation 2013-03-22 14:31:31
Last modified on 2013-03-22 14:31:31
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 8
Author CWoo (3771)
Entry type Definition
Classification msc 15A09
Classification msc 60J10
Synonym Moore-Penrose pseudoinverseMathworldPlanetmath
Related topic DrazinInverse
Related topic Pseudoinverse