{"paper":{"title":"On Grioli's minimum property and its relation to Cauchy's polar decomposition","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Angela Madeo, Johannes Lankeit, Patrizio Neff","submitted_at":"2013-09-30T13:12:09Z","abstract_excerpt":"The unitary polar factor of a matrix F is the unitary matrix Q realizing the minimum of the norm of F-Q over all unitary matrices Q. Tracing back the development on the optimality of the polar factor to its presumable roots, in this paper we present a commented translation of a note by G. Grioli from 1940 (Boll.Un.Math.Ital.2,452-455(1940)) showing the minimization property for the Frobenius norm in dimension 3; in his words:\n  \"We show that for the homogeneous displacement, tangent to any finite displacement, there exists a minimum property completely analogous to a well known property valid "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1310.7826","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"}