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
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
See also
Drazin Inverse, Least Squares Fitting, Matrix Inverse, PseudoinverseExplore with Wolfram|Alpha
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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 InverseCite this as:
Weisstein, Eric W. "Moore-Penrose Matrix Inverse." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Moore-PenroseMatrixInverse.html