Minimal properties of Moore-Penrose inverses

AbstractLet Ax = y be consistent; let x0 = Gy be any minimum-norm solution satisfying (AG)′ = AG; and let A+ be the Moore-Penrose inverse of A. It is shown that φ(G) ⩾ φ(A+) for any φ in a class Φ containing the unitarily invariant matrix norms. The conditioning of the system Ax = y is studied via condition numbers Cφ(A, G). It is shown that Cφ(A, G) ⩾ Cφ(A, A+) for every φ∈ Φ. Moreover, bounds on Cφ(A, G) 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