A matrix lower bound

Four essentially different interpretations of a lower bound for linear operators are shown to be equivalent for matrices (involving inequalities, convex sets, minimax problems, and quotient spaces). Properties stated by von Neumann in a restricted case are satisfied by the lower bound. Applications are made to rank reduction, s-numbers, condition numbers, and pseudospectra. In particular, the matrix lower bound is the distance to the nearest matrix with strictly contained row or column spaces, and it occurs in a condition number formula for any consistent system of linear equations, including those that are underdetermined.