{"paper":{"title":"Matrix Vieta Theorem","license":"","headline":"","cross_cats":["math.OA"],"primary_cat":"math.RA","authors_text":"Albert Schwarz, Dmitry Fuchs","submitted_at":"1994-10-25T00:00:00Z","abstract_excerpt":"We consider generalizations of the Vieta formula (relating the coefficients of an algebraic equation to the roots) to the case of equations whose coefficients are order-$k$ matrices.\n  Specifically, we prove that if $X_1,\\dots ,X_n$ are solutions of an algebraic matrix equation $X^n+A_1X^{n-1}+\\dots +A_n=0,$ independent in the sense that they determine the coefficients $A_1,\\dots ,A_n$, then the trace of $A_1$ is the sum of the traces of the $X_i$, and the determinant of $A_n$ is, up to a sign, the product of the determinants of the $X_i$. We generalize this to arbitrary rings with appropriate"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/9410207","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"}