Given an matrix , the Moore-Penrose generalized matrix
inverse is a unique matrix pseudoinverse .
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
is the conjugate transpose .
It is also true that
(5)
is the shortest length least squares solution
to the problem
(6)
If the inverse of exists, then
(7)
as can be seen by premultiplying both sides of (6 ) by to create a square matrix
which can then be inverted,
(8)
giving
See also Drazin Inverse ,
Least
Squares Fitting ,
Matrix Inverse ,
Pseudoinverse
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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 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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