  The well-known M-P (Moore-Penrose) pseudoinverse is used in several
linear-algebra applications; for example, to compute least-squares solutions of
inconsistent systems of linear equations. It is uniquely characterized by four
properties, but not all of them need to be satisfied for some applications. For
computational reasons, it is convenient then, to construct sparse
block-structured matrices satisfying relevant properties of the M-P
pseudoinverse for specific applications. (Vector) 1-norm minimization has been
used to induce sparsity in this context. Aiming at row-sparse generalized
inverses motivated by the least-squares application, we consider 2,1-norm
minimization (and generalizations). In particular, we show that a 2,1-norm
minimizing generalized inverse satisfies two additional M-P pseudoinverse
properties, including the one needed for computing least-squares solutions. We
present mathematical-optimization formulations related to finding row-sparse
generalized inverses that can be solved very efficiently, and compare their
solutions numerically to generalized inverses constructed by other
methodologies, also aiming at sparsity and row-sparse structure.
