{"doi":"10.1090/proc/14831","title":"On matrix rearrangement inequalities","abstract":"Given two symmetric and positive semidefinite square matrices <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper A comma upper B\"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>,</mml:mo> <mml:mi>B</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">A, B</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , is it true that any matrix given as the product of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"m\"> <mml:semantics> <mml:mi>m</mml:mi> <mml:annotation encoding=\"application/x-tex\">m</mml:annotation> </mml:semantics> </mml:math> </inline-formula> copies of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper A\"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding=\"application/x-tex\">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n\"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding=\"application/x-tex\">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> copies of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper B\"> <mml:semantics> <mml:mi>B</mml:mi> <mml:annotation encoding=\"application/x-tex\">B</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in a particular sequence must be dominated in the spectral norm by the ordered matrix product <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper A Superscript m Baseline upper B Superscript n\"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>A</mml:mi> <mml:mi>m</mml:mi> </mml:msup> <mml:msup> <mml:mi>B</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">A^m B^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> ? For example, is <disp-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-vertical-bar upper A upper A upper B upper A upper A upper B upper A upper B upper B double-vertical-bar less-than-or-equal-to double-vertical-bar upper A upper A upper A upper A upper A upper B upper B upper B upper B double-vertical-bar question-mark\"> <mml:semantics> <mml:mrow> <mml:mo fence=\"false\" stretchy=\"false\"> ‖ </mml:mo> <mml:mi>A</mml:mi> <mml:mi>A</mml:mi> <mml:mi>B</mml:mi> <mml:mi>A</mml:mi> <mml:mi>A</mml:mi> <mml:mi>B</mml:mi> <mml:mi>A</mml:mi> <mml:mi>B</mml:mi> <mml:mi>B</mml:mi> <mml:mo fence=\"false\" stretchy=\"false\"> ‖ </mml:mo> <mml:mo> ≤ </mml:mo> <mml:mo fence=\"false\" stretchy=\"false\"> ‖ </mml:mo> <mml:mi>A</mml:mi> <mml:mi>A</mml:mi> <mml:mi>A</mml:mi> <mml:mi>A</mml:mi> <mml:mi>A</mml:mi> <mml:mi>B</mml:mi> <mml:mi>B</mml:mi> <mml:mi>B</mml:mi> <mml:mi>B</mml:mi> <mml:mo fence=\"false\" stretchy=\"false\"> ‖ </mml:mo> <mml:mo>?</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">\\begin{equation*} \\| AABAABABB \\| \\leq \\| AAAAABBBB \\| ? \\end{equation*}</mml:annotation> </mml:semantics> </mml:math> </disp-formula> 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 <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper A comma upper B\"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>,</mml:mo> <mml:mi>B</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">A,B</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . However, the <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"1\"> <mml:semantics> <mml:mn>1</mml:mn> <mml:annotation encoding=\"application/x-tex\"","journal":"Proceedings of the American Mathematical Society","year":2020,"id":128617,"datarank":0.0,"base_score":0.0,"endowment":0.0,"self_citation_contribution":0.0,"citation_network_contribution":0.0,"self_endowment_contribution":0.0,"citer_contribution":0.0,"corpus_percentile":null,"corpus_rank":null,"citation_count":0,"citer_count":0,"citers_with_citation_signal":0,"citers_with_endowment":0,"datacite_reuse_total":0,"is_dataset":false,"is_dataset_confidence":0.9362,"is_data_producer":false,"deposit_databanks":null,"is_oa":true,"file_count":0,"downloads":0,"has_version_chain":false,"published_date":"2020-01-01","fair_score":null,"fair_percentile":null,"algorithm_id":"datarank_citation_only_1hop_v6","ranking_scope":"data_only","authors":[{"id":416446,"name":"Xiuyuan Cheng","orcid":"0000-0002-1034-6019","position":1,"is_corresponding":false},{"id":579320,"name":"Lillian B. Pierce","orcid":"0000-0002-0194-0083","position":2,"is_corresponding":false},{"id":579321,"name":"Stefan Steinerberger","orcid":"0000-0002-7745-4217","position":3,"is_corresponding":false},{"id":579319,"name":"Rima Alaifari","orcid":"0000-0003-1608-8580","position":0,"is_corresponding":true}],"reference_count":31,"raw_metadata":null,"created_at":"2026-07-18T23:15:42.522809Z","pmid":null,"pmcid":null,"fwci":null,"citation_percentile":null,"influential_citations":0,"oa_status":null,"license":null,"views":0,"total_file_size_bytes":0,"version_count":0,"fair_f":null,"fair_a":null,"fair_i":null,"fair_r":null,"fair_zscore":null,"fair_rationale":null,"fair_model":null,"fair_agent_version":null,"fair_fulltext_source":null,"fair_has_llm":null,"fair_computed_at":null,"clinical_trials":[],"software_tools":[],"db_accessions":[],"linked_datasets":[],"topics":[]}