Given an matrix matrix
inverse is a unique matrix pseudoinverse matrix 1-inverse , and is implemented in the
Wolfram Language as PseudoInverse m ].
The Moore-Penrose inverse satisfies
where conjugate transpose .
It is also true that
(5)
is the shortest length least squares solution
to the problem
(6)
If the inverse of
(7)
as can be seen by premultiplying both sides of (6 ) by 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. Campbell,
S. L. and Meyer, C. D. Jr. Generalized
Inverses of Linear Transformations. Lawson,
C. and Hanson, R. Solving
Least Squares Problems. 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. Referenced
on Wolfram|Alpha Moore-Penrose Matrix Inverse
Cite this as:
Weisstein, Eric W. "Moore-Penrose Matrix Inverse."
From MathWorld https://mathworld.wolfram.com/Moore-PenroseMatrixInverse.html
Subject classifications
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].
is the conjugate transpose.
exists, then
to create a square matrix
which can then be inverted,
