A norm compression inequality for block partitioned positive semidefinite matrices

Let A be a positive semidefinite matrix, block partitioned as A = ((B)(C*) (C)(D)), where B and D are square blocks. We prove the following inequalities for the Schatten q-norm parallel to(.)parallel to q: parallel to A parallel to(q)(q) <= (2(q) -2) parallel to C parallel to(q)(q) + parallel to B parallel to(q)(q) + parallel to D parallel to(q)(q), 1 <= q <= 2, and parallel to A parallel to(q)(q) >= (2(q) - 2) parallel to C parallel to(q)(q) + parallel to B parallel to(q)(q) + parallel to D parallel to(q)(q), 2 <= q. We show that these bounds obey a strong sharpness condition when the blocks are of size at least 2 x 2, and parallel to B parallel to(q), parallel to D parallel to(q) >= parallel to C parallel to(q). Finally, our bounds can be extended to symmetric partitionings into larger numbers of blocks: for A = [A(ij)], parallel to A parallel to(q)(q) <= Sigma(i) parallel to A(ii)parallel to(q)(q) + (2(q) - 2) Sigma(i < j) parallel to A(ij)parallel to(q)(q), 1 <= q <= 2, while for 2 <= q the inequality is reversed. (c) 2005 Elsevier Inc. All rights reserved.