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Moore-Penrose Generalized Matrix Inverse

Given an $m\times n$ Matrix ${\hbox{\sf B}}$, the Moore-Penrose generalized Matrix Inverse is a unique $n\times m$ Matrix ${\hbox{\sf B}}^+$ which satisfies

$\displaystyle {\hbox{\sf B}}{\hbox{\sf B}}^+{\hbox{\sf B}}$ $\textstyle =$ $\displaystyle {\hbox{\sf B}}$ (1)
$\displaystyle {\hbox{\sf B}}^+{\hbox{\sf B}}{\hbox{\sf B}}^+$ $\textstyle =$ $\displaystyle {\hbox{\sf B}}^+$ (2)
$\displaystyle ({\hbox{\sf B}}{\hbox{\sf B}}^+)^{\rm T}$ $\textstyle =$ $\displaystyle {\hbox{\sf B}}{\hbox{\sf B}}^+$ (3)
$\displaystyle ({\hbox{\sf B}}^+{\hbox{\sf B}})^{\rm T}$ $\textstyle =$ $\displaystyle {\hbox{\sf B}}^+{\hbox{\sf B}}.$ (4)

It is also true that
{\bf z}={\hbox{\sf B}}^+{\bf c}
\end{displaymath} (5)

is the shortest length Least Squares solution to the problem
{\hbox{\sf B}}{\bf z}={\bf c}.
\end{displaymath} (6)

If the inverse of $({\hbox{\sf B}}^{\rm T}{\hbox{\sf B}})$ exists, then

{\hbox{\sf B}}^+ = ({\hbox{\sf B}}^{\rm T}{\hbox{\sf B}})^{-1}{\hbox{\sf B}}^{\rm T},
\end{displaymath} (7)

where ${\hbox{\sf B}}^{\rm T}$ is the Matrix Transpose, as can be seen by premultiplying both sides of (7) by ${\hbox{\sf B}}^{\rm T}$ to create a Square Matrix which can then be inverted,
{\hbox{\sf B}}^{\rm T}{\hbox{\sf B}}{\bf z}={\hbox{\sf B}}^{\rm T}{\bf c},
\end{displaymath} (8)

$\displaystyle {\bf z}$ $\textstyle =$ $\displaystyle ({\hbox{\sf B}}^{\rm T}{\hbox{\sf B}})^{-1}{\hbox{\sf B}}^{\rm T}{\bf c}$  
  $\textstyle \equiv$ $\displaystyle {\hbox{\sf B}}^+{\bf c}.$ (9)

See also Least Squares Fitting, Matrix Inverse


Ben-Israel, A. and Greville, T. N. E. Generalized Inverses: Theory and Applications. New York: Wiley, 1977.

Lawson, C. and Hanson, R. Solving Least Squares Problems. Englewood Cliffs, NJ: Prentice-Hall, 1974.

Penrose, R. ``A Generalized Inverse for Matrices.'' Proc. Cambridge Phil. Soc. 51, 406-413, 1955.

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© 1996-9 Eric W. Weisstein