{"paper":{"title":"Trace and determinant preserving maps of matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.RA","authors_text":"Chih-Neng Liu, Huajun Huang, Jun Zhang, Ming-Cheng Tsai, Patricia Szokol","submitted_at":"2016-03-12T07:52:19Z","abstract_excerpt":"Suppose a map $\\phi$ on the set of positive definite matrices satisfies $\\det(A+B)=\\det(\\phi(A)+\\phi(B))$. Then we have $${\\rm tr}(AB^{-1}) = {\\rm tr}(\\phi(A){\\phi(B)}^{-1}).$$ Through this viewpoint, we show that $\\phi$ is of the form $\\phi(A)= M^*AM$ or $\\phi(A)= M^*A^tM$ for some invertible matrix $M$ with $\\det (M^*M)=1$. We also characterize the map $\\phi: \\mathcal{S} \\rightarrow \\mathcal{S}$ preserving the determinant of convex combinations in $\\mathcal{S}$ by using similar method. Here $\\mathcal{S}$ can be the set of complex matrices, positive definite matrices, symmetric matrices, and "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.03869","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"}