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Date: 18-2-2021
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An matrix is a 1-inverse of an matrix for which
(1) |
The Moore-Penrose matrix inverse is a particular type of 1-inverse.
A matrix equation
(2) |
has a solution iff
(3) |
(Campbell and Meyer 1991).
Let be an matrix and use elementary row operations (through premultiplication by a nonsingular matrix obtained by performing the same operations on the identity matrix) and elementary column operations (through postmultiplication by a nonsingular matrix obtained by performing the same operations on the identity matrix) to transform into the form
(4) |
where is the block matrix
(5) |
and is an identity matrix with the rank of . Then a matrix is a 1-inverse of iff there are appropriately dimensional matrices , and such that
(6) |
(Jodár et al. 1991).
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.
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