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

$${\hbox{\sf B}}{\hbox{\sf B}}^+{\hbox{\sf B}} = {\hbox{\sf B}}$$ (1)

$${\hbox{\sf B}}^+{\hbox{\sf B}}{\hbox{\sf B}}^+ = {\hbox{\sf B}}^+$$ (2)

$$({\hbox{\sf B}}{\hbox{\sf B}}^+)^{\rm T} = {\hbox{\sf B}}{\hbox{\sf B}}^+$$ (3)

$$({\hbox{\sf B}}^+{\hbox{\sf B}})^{\rm T} = {\hbox{\sf B}}^+{\hbox{\sf B}}.$$ (4)

It is also true that

$${\bf z}={\hbox{\sf B}}^+{\bf c}$$ (5)

is the shortest length Least Squares solution to the problem

$${\hbox{\sf B}}{\bf z}={\bf c}.$$ (6)

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

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

$${\hbox{\sf B}}^{\rm T}{\hbox{\sf B}}{\bf z}={\hbox{\sf B}}^{\rm T}{\bf c},$$ (8)

giving

$${\bf z} = ({\hbox{\sf B}}^{\rm T}{\hbox{\sf B}})^{-1}{\hbox{\sf B}}^{\rm T}{\bf c}$$

$$\equiv {\hbox{\sf B}}^+{\bf c}.$$ (9)

See also Least Squares Fitting, Matrix Inverse

References

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.