{"paper":{"title":"Weak log majorization and determinantal inequalities","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Pingping Zhang, Tin-Yau Tam","submitted_at":"2016-11-16T01:04:22Z","abstract_excerpt":"Denote by $\\P_n$ the set of $n\\times n$ positive definite matrices. Let $D = D_1\\oplus \\dots \\oplus D_k$, where $D_1\\in \\P_{n_1}, \\dots, D_k \\in \\P_{n_k}$ with $n_1+\\cdots + n_k=n$. Partition $C\\in \\P_n$ according to $(n_1, \\dots, n_k)$ so that $\\Diag C = C_1\\oplus \\dots \\oplus C_k$. We prove the following weak log majorization result: \\begin{equation*} \\lambda (C^{-1}_1D_1\\oplus \\cdots \\oplus C^{-1}_kD_k)\\prec_{w \\,\\log} \\lambda(C^{-1}D), \\end{equation*} where $\\lambda(A)$ denotes the vector of eigenvalues of $A\\in \\Cnn$. The inequality does not hold if one replaces the vectors of eigenvalues"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.05108","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"}