# 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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