On matrix rearrangement inequalities

Given two symmetric and positive semidefinite square matrices A,B, is it true that any matrix given as the product of m copies of A and n copies of B in a particular sequence must be dominated in the spectral norm by the ordered matrix product AmBn? For example, is ∥ AABAABABB ∥ ≤ ∥AAAAABBBB∥? Drury [Electron J. Linear Algebra 18 (2009), pp. 13 20] has characterized precisely which disordered words have the property that an inequality of this type holds for all matrices A,B. However, the 1-parameter family of counterexamples Drury constructs for these characterizations is comprised of 3×3 matrices, and thus as stated the characterization applies only for N × N matrices with N ≤ 3. In contrast, we prove that for 2 × 2 matrices, the general rearrangement inequality holds for all disordered words. We also show that for larger N ×N matrices, the general rearrangement inequality holds for all disordered words for most A,B (in a sense of full measure) that are sufficiently small perturbations of the identity.

Duke Scholars

Author Xiuyuan Cheng Mathematics

Author Lillian Beatrix Pierce Mathematics