Moore-Penrose Matrix Inverse

Author

Eric W. Weisstein
December 16, 2025

This notebook downloaded from https://mathworld.wolfram.com/notebooks/LinearAlgebra/Moore-PenroseMatrixInverse.nb.

For more information, see Eric's MathWorld entry https://mathworld.wolfram.com/Moore-PenroseMatrixInverse.html.

©2025 Wolfram Research, Inc. except for portions noted otherwise

Example

h={{1,0,0,1},{0,1,I,0},{0,-I,1,0},{1,0,0,1}};

hplus=PseudoInverse[h];

h.hplus.h==h

hplus.h.hplus==hplus

ConjugateTranspose[h.hplus]==h.hplus

ConjugateTranspose[hplus.h]==hplus.h

hplus==Inverse[ConjugateTranspose[hplus].h].ConjugateTranspose[h]