A Norm Compression Inequality for Block Partitioned Positive Semidefinite Matrices

Abstract

Let $A$ be a positive semidefinite matrix, block partitioned as $$ A=\twomat{B}{C}{C^*}{D}, $$ where $B$ and $D$ are square blocks. We prove the following inequalities for the Schatten $q$-norm $||.|{|}_q$, which are sharp when the blocks are of size at least $2\times 2$: $$||A|{|}_q^q\le (2^q-2)||C|{|}_q^q+||B|{|}_q^q+||D|{|}_q^q,1\le q\le 2,$$ and $$||A|{|}_q^q\ge (2^q-2)||C|{|}_q^q+||B|{|}_q^q+||D|{|}_q^q,2\le q.$$ These bounds can be extended to symmetric partitionings into larger numbers of blocks, at the expense of no longer being sharp: $$||A|{|}_q^q\le \sum _i||A_{ii}|{|}_q^q+(2^q-2)\sum _{i<j}||A_{ij}|{|}_q^q,1\le q\le 2,$$ and $$||A|{|}_q^q\ge \sum _i||A_{ii}|{|}_q^q+(2^q-2)\sum _{i<j}||A_{ij}|{|}_q^q,2\le q.$$

24 pages