A generalization of the Moore-Penrose inverse related to matrix subspaces of Cn×m

A natural generalization of the classical Moore-Penrose inverse is presented. The so-called S-Moore-Penrose inverse of an m×n complex matrix A, denoted by AS†, is defined for any linear subspace S of the matrix vector space Cn×m. The S-Moore-Penrose inverse AS† is characterized using either the singular value decomposition or (for the full rank case) the orthogonal complements with respect to the Frobenius inner product. These results are applied to the preconditioning of linear systems based on Frobenius norm minimization and to the linearly constrained linear least squares problem.