This is a generalization of the original concept of Moore-Penrose inverse (MPI). The weighted MPI of a matrix is defined by the following four properties:
- (A): ,
- (B): ,
- (C)_N: ,
- (D)_M: .
where the weighting matrices and are of the orders and .
When weighting matrices are equal to the corresponding identities, the above definition reduces to ordinary MPI .
Source: R. B. Bapat, S. K. Jain and S. Pati “Weighted Moore-Penrose Inverse of a Boolean Matrix”; Linear Algebra and Its Applications 225:267-279 (1997); North-Holland; pg. 692-704. http://www.math.ohiou.edu/~jain/077.pdf.
More at this blog:
- Some examples of the Moore-Penrose inverse – https://eikonal.wordpress.com/2010/02/17/some-examples-of-the-moore-penrose-inverse/
- Moore-Penrose inverse for light-cone vectors – https://eikonal.wordpress.com/2010/02/26/moore-perose-inverse-for-light-cone-vectors/