Structures and uniqueness conditions of \(MK\)-weighted pseudoinverses
For given matrices \(M\), \(A\) and \(K\), two kinds of \(MK\)-weighted pseudoinverses \(X\) of \(A\) are studied. The first kind is characterized by the following four conditions (similar to the Moore-Penrose conditions): \(MAXA = MA\), \(XAX = X\), \((M^ HMAX)^ H =M^ H MAX\), \((K^ H KXA)^ H =K^ HKXA\). The second kind is characterized by \[ \min_{X \in S} \| X \|_ K \quad \text{with} \quad S = \biggl\{ X : \| AX - I \|_ M = \min_{W \in C^{n \times s}} \| AW - I \|_ M \biggr\}. \] Structures of these \(MK\)-weighted pseudoinverses of \(A\) are deduced, and uniqueness conditions are obtained.
scientific article; zbMATH DE number 698684
scientific article