Minimal properties of Moore-Penrose inverses (Q1316195)

Let \(Ax=y\) be consistent and let \(x_ 0=Gy\) be any minimum-norm solution satisfying \((AG)^ T=AG\). Let \(A^ +\) be the Moore-Penrose inverse of \(A\). It is shown that \(\varphi(G) \geq \varphi(A^ +)\) for any \(\varphi\) in a class \(\Phi\) containing the unitarily invariant matrix norms. The conditioning of the system \(Ax=y\) is studied via condition numbers \(C_ \varphi (A,G)\). It is shown that \(C_ \varphi (A,G) \geq C_ \varphi (A,A^ +)\) for every \(\varphi \in \Phi\). Bounds are given in terms of singular values. Parallel results are found when \(A\) and \(G\) are symmetric with applications to linear models of less than full rank.