The best generalized inverse
If \(A\) is an \(m\times n\) matrix, an approximate solution to the problem \(Ax=y\), is \(x=By\), where \(B\) is an \(n\times m\) matrix, and it is called a generalized inverse of \(A\). The Moore-Penrose generalized inverse of \(A\) (\(\widetilde{A}=(|A|_{\mid\text{range}(|A|)})^{-1}U^{\ast }\), where \(|A|=(A^{\ast }A)^{1/2}\) is the absolute value of \(A\) and \(U\) is the linear and isometric map from \(|A|x\) to \(Ax\), as a function from \(\text{range}(|A|)\) to \(\text{range}(A)\)) minimizes \(\|Ax-y\|^{2}=\|(AB-I)y\|^{2}\) where \(\|\cdot \|\) is the Euclidean norm\(.\) The main result of the paper is that for all norms type ``Schatten-von Neumann'' the Moore-Penrose generalized inverse \(\widetilde{A}\) is best, that is, for any \(p\), \(1\leq p\leq \infty \), and for any \(n\times m\) matrix \(B\), \(\|A\widetilde{A}-I\|_{p}\leq \|AB-I\|_{p}\). This result is also established to the more general context of all unitarily invariant norms \(|||\cdot |||\), that is, \(\min\{|||AB-I|||\mid B\in \mathfrak{M}_{n\times m}\}=|||A\widetilde{A}-I|||\) for every unitarily invariant norm (a norm \(|||\cdot |||\) is called unitarily invariant if \(|||UAV|||\) \(=|||A|||\) for all \(A\in \mathfrak{M}_{m\times n}\) and all unitary operators \(U\), \(V\) in \( \mathfrak{M}_{m\times m}\), \(\mathfrak{M}_{n\times n}\), respectively).
scientific article; zbMATH DE number 1563450