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Moore-Penrose Matrix Inverse


Given an m×n matrix B, the Moore-Penrose generalized matrix inverse is a unique n×m matrix pseudoinverse B^+. This matrix was independently defined by Moore in 1920 and Penrose (1955), and variously known as the generalized inverse, pseudoinverse, or Moore-Penrose inverse. It is a matrix 1-inverse, and is implemented in the Wolfram Language as PseudoInverse[m].

The Moore-Penrose inverse satisfies

BB^+B=B
(1)
B^+BB^+=B^+
(2)
(BB^+)^(H)=BB^+
(3)
(B^+B)^(H)=B^+B,
(4)

where B^(H) is the conjugate transpose.

It is also true that

 z=B^+c
(5)

is the shortest length least squares solution to the problem

 Bz=c.
(6)

If the inverse of (B^(H)B) exists, then

 B^+=(B^(H)B)^(-1)B^(H),
(7)

as can be seen by premultiplying both sides of (6) by B^(H) to create a square matrix which can then be inverted,

 B^(H)Bz=B^(H)c,
(8)

giving

z=(B^(H)B)^(-1)B^(H)c
(9)
=B^+c.
(10)

See also

Drazin Inverse, Least Squares Fitting, Matrix Inverse, Pseudoinverse

Explore with Wolfram|Alpha

References

Ben-Israel, A. and Greville, T. N. E. Generalized Inverses: Theory and Applications. New York: Wiley, 1977.Campbell, S. L. and Meyer, C. D. Jr. Generalized Inverses of Linear Transformations. New York: Dover, 1991.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.Rao, C. R. and Mitra, S. K. Generalized Inverse of Matrices and Its Applications. New York: Wiley, 1971.

Referenced on Wolfram|Alpha

Moore-Penrose Matrix Inverse

Cite this as:

Weisstein, Eric W. "Moore-Penrose Matrix Inverse." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Moore-PenroseMatrixInverse.html

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