On norms of principal submatrices

Abstract

Let a norm on the set Mn of real or complex n-by-n matrices be given. We investigate the question of finding the largest constants αn and βn such that for each A∈Mn the average of the norms of its (n−1)-by-(n−1) principal submatrices is at least αn times the norm of A, and such that the maximum of the norms of those principal submatrices is at least βn times the norm of A. For a variety of classical norms including induced ℓp-norms, weakly unitarily invariant norms, and entrywise norms we give lower and upper bounds for αn and βn. In several cases αn and βn are explicitly determined.