The inversePlanetmathPlanetmath A-1 of a matrix A exists only if A is square and has full rank. In this case, Ax=b has the solution x=A-1b.

The pseudoinverseMathworldPlanetmath A+ (beware, it is often denoted otherwise) is a generalization of the inverse, and exists for any m×n matrix. We assume m>n. If A has full rank (n) we define:


and the solution of Ax=b is x=A+b.

More accurately, the above is called the Moore-Penrose pseudoinverseMathworldPlanetmath.

1 Calculation

The best way to compute A+ is to use singular value decompositionMathworldPlanetmath. With A=USVT , where U and V (both n×n) orthogonalMathworldPlanetmathPlanetmath and S (m×n) is diagonalMathworldPlanetmath with real, non-negative singular values σi, i=1,,n. We find


If the rank r of A is smaller than n, the inverse of STS does not exist, and one uses only the first r singular values; S then becomes an r×r matrix and U,V shrink accordingly. see also Linear Equations.

2 Generalization

The term “pseudoinverse” is actually used for any operator pinv satisfying


for a m×n matrix M. Beyond this, pseudoinverses can be defined on any reasonable matrix identity.


  • Originally from The Data Analysis Briefbook (

Title pseudoinverse
Canonical name Pseudoinverse
Date of creation 2013-03-22 12:07:21
Last modified on 2013-03-22 12:07:21
Owner akrowne (2)
Last modified by akrowne (2)
Numerical id 6
Author akrowne (2)
Entry type Definition
Classification msc 15-00
Classification msc 65-00
Synonym pseudo-inverse
Synonym Moore-Penrose pseudoinverse
Related topic MoorePenroseGeneralizedInverse