{"paper":{"title":"How many matrices can be spectrally balanced simultaneously?","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.FA","authors_text":"Fedor Nazarov, Ronen Eldan, Yuval Peres","submitted_at":"2016-06-06T10:23:29Z","abstract_excerpt":"We prove that any $\\ell$ positive definite $d \\times d$ matrices, $M_1,\\ldots,M_\\ell$, of full rank, can be simultaneously spectrally balanced in the following sense: for any $k < d$ such that $\\ell \\leq \\lfloor \\frac{d-1}{k-1} \\rfloor$, there exists a matrix $A$ satisfying $\\frac{\\lambda_1(A^T M_i A) }{ \\mathrm{Tr}( A^T M_i A ) } < \\frac{1}{k}$ for all $i$, where $\\lambda_1(M)$ denotes the largest eigenvalue of a matrix $M$. This answers a question posed by Peres, Popov and Sousi and completes the picture described in that paper regarding sufficient conditions for transience of self-interacti"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.01680","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}