BIT Numerical Mathematics

, Volume 34, Issue 3, pp 437–450

# Structures and uniqueness conditions of MK-weighted pseudoinverses

• Musheng Wei
• Birong Zhang
Article

## Abstract

For given matricesM, A andK of appropriate sizes, we study the following two kinds of MK-weighted pseudoinverses ofA. The first kind of weighted pseudoinverseX is characterized by the following four Moore-Penrose like conditions:
$$\begin{gathered} (1)_M MAXA = MA; \hfill \\ (2) XAX = X; \hfill \\ (3)_M \left( {M^H MAX} \right)^H = M^H MAX; \hfill \\ (4)_K \left( {K^H KXA} \right)^H = K^H KXA. \hfill \\ \end{gathered}$$
The second kind of weighted pseudoinverseX is characterized by
$$\mathop {\min }\limits_{X \in S} \parallel X\parallel _K , with S = \{ X: \parallel AX - I\parallel _M = \mathop {\min }\limits_{W \in C^{n \times s} } \parallel AW - I\parallel _M \} .$$
Structures of these MK-weighted pseudoinverses ofA are deduced, and the uniqueness conditions of such pseudoinverses are obtained.

15A09

### Key words

Weighted pseudoinverse constrained least squares uniqueness

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