Matrix 1-Inverse

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.