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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.

A matrix equation

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

J=[I 0; 0 0]
(5)

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

A^-=Q[I X; Y Z]P
(6)

(Jodár et al. 1991).

See also

Drazin Inverse, Matrix Inverse, Moore-Penrose Matrix Inverse, Pseudoinverse

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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.

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Matrix 1-Inverse

Cite this as:

Weisstein, Eric W. "Matrix 1-Inverse." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Matrix1-Inverse.html

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