{"paper":{"title":"A note on the $\\top$-Stein matrix equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Chun-Yueh Chiang","submitted_at":"2013-05-18T08:39:39Z","abstract_excerpt":"This note is concerned with the linear matrix equation $X = AX^\\top B + C$, where the operator $(\\cdot)^\\top$ denotes the transpose ($\\top$) of a matrix. The first part of this paper set forth the necessary and sufficient conditions for the unique solvability of the solution $X$. The second part of this paper aims to provide a comprehensive treatment of the relationship between the theory of the generalized eigenvalue problem and the theory of the linear matrix equation. In the finally part of this paper starts with a briefly review of numerical methods for solving the linear matrix equation. "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1305.4243","kind":"arxiv","version":3},"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"}