On norms of principal submatrices

Let a norm on the set M-n of real or complex n-by-nmatrices be given. We investigate the question of finding the largest constants alpha(n) and beta(n) such that for each A is an element of M(n)the average of the norms of its (n-1)-by-(n-1) principal submatrices is at least alpha(n) times the norm ofA, and such that the maximum of the norms of those principal submatrices is at least beta(n) times the norm of A. For a variety of classical norms including induced l(p)-norms, weakly unitarily invariant norms, and entrywise norms we give lower and upper bounds for alpha(n) and beta(n). In several cases anand beta(n) are explicitly determined. (C) 2021 Elsevier Inc. All rights reserved.