{"paper":{"title":"Sum of squared logarithms - An inequality relating positive definite matrices and their matrix logarithm","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Johannes Lankeit, Mircea Birsan, Patrizio Neff","submitted_at":"2013-01-24T12:59:00Z","abstract_excerpt":"Let y1, y2, y3, a1, a2, a3 > 0 be such that y1 y2 y3 = a1 a2 a3 and\ny1 + y2 + y3 >= a1 + a2 + a3,\ny1 y2 + y2 y3 + y1 y3 >= a1 a2 + a2 a3 + a1 a3.\n  Then the following inequality holds\n(log y1)^2 + (log y2)^2 + (log y3)^2 >= (log a1)^2 + (log a2)^2 + (log a3)^2.\n  This can also be stated in terms of real positive definite 3x3-matrices P1, P2: If their determinants are equal det P1 = det P2, then\ntr P1 >= tr P2 and tr Cof P1 >= tr Cof P2 implies norm(log P1) >= norm(log P2),\nwhere log is the principal matrix logarithm and norm(P) denotes the Frobenius matrix norm. Applications in matrix analysis"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1301.6604","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"}