Eikonal Blog

2010.02.19

Weighted Moore-Penrose inverse

Filed under: mathematics — Tags: — sandokan65 @ 15:13

This is a generalization of the original concept of Moore-Penrose inverse (MPI). The weighted MPI A^{+(N,M)} of a matrix A \in {\Bbb F}^{n\times m} is defined by the following four properties:

  • (A): A \cdot A^{+(N,M)} \cdot A = A,
  • (B): A^{+(N,M)} \cdot A \cdot A^{+(N,M)} = A^{+(N,M)},
  • (C)_N: (M \cdot A \cdot A^{+(N,M)})^c  = M \cdot A \cdot A^{+(N,M)},
  • (D)_M: (A^{+(N,M)}\cdot A \cdot N)^c  = A^{+(N,M)}\cdot A \cdot N.

where the weighting matrices M and N are of the orders n\times n and m\times m.

When weighting matrices are equal to the corresponding identities, the above definition reduces to ordinary MPI A^c.

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.


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