Matrix 1-Inverse
An n×m matrix A^- is a 1-inverse of an m×n matrix A for which
| AA^-A=A. |
(1)
|
The Moore-Penrose matrix inverse is a particular type of 1-inverse.
| Ax=b |
(2)
|
has a solution iff
| AA^-b=b |
(3)
|
(Campbell and Meyer 1991).
Let A be an m×n matrix and use elementary row operations (through premultiplication by a nonsingular matrix P obtained by performing the same operations on the identity matrix) and elementary column operations (through postmultiplication by a nonsingular matrix Q obtained by performing the same operations on the identity matrix) to transform A into the form
| PAQ=J, |
(4)
|
where J is the block matrix
| [画像: J=[I 0; 0 0] ] |
(5)
|
![画像: J=[I 0; 0 0]](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/Matrix1-Inverse/NumberedEquation5.svg){kind=link}
and I is an r×r identity matrix with r the rank of A. Then a matrix A^- is a 1-inverse of A iff there are appropriately dimensional matrices X, Y and Z such that
| [画像: A^-=Q[I X; Y Z]P ] |
(6)
|
![画像: A^-=Q[I X; Y Z]P](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/Matrix1-Inverse/NumberedEquation6.svg){kind=link}
(Jodár et al. 1991).
See also
Drazin Inverse, Matrix Inverse, Moore-Penrose Matrix Inverse, PseudoinverseExplore with Wolfram|Alpha
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References
Campbell, S. L. and Meyer, C. D. Jr. Generalized Inverses of Linear Transformations. New York: Dover, 1991.Jodár, L.; Law, A. G.; Rezazadeh, A.; Watson, J. H.; and Wu, G. "Computations for the Moore-Penrose and Other Generalized Inverses." Congress. Numer. 80, 57-64, 1991.Rao, C. R. and Mitra, S. K. Generalized Inverse of Matrices and Its Applications. New York: Wiley, 1971.Referenced on Wolfram|Alpha
Matrix 1-InverseCite this as:
Weisstein, Eric W. "Matrix 1-Inverse." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Matrix1-Inverse.html