{"paper":{"title":"Roth's solvability criteria for the matrix equations ${AX-\\widehat XB=C}$ and ${X-A\\widehat{X}B=C}$ over the skew field of quaternions with an involutive automorphism $q\\mapsto \\hat q$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RT"],"primary_cat":"math.RA","authors_text":"Tetiana Klymchuk, Vladimir V. Sergeichuk, Vyacheslav Futorny","submitted_at":"2016-11-11T00:37:53Z","abstract_excerpt":"The matrix equation $AX-XB=C$ has a solution if and only if the matrices [A&C\\\\0&B] and [A &0\\\\0 & B] are similar. This criterion was proved over a field by W.E. Roth (1952) and over the skew field of quaternions by Huang Liping (1996). H.K. Wimmer (1988) obtained an analogous criterion for the matrix equation $X-AXB=C$ over a field. We extend these criteria to the matrix equations $AX-\\widehat XB=C$ and $X-A\\widehat XB=C$ over the skew field of quaternions with a fixed involutive automorphism $q\\mapsto \\hat q$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.04527","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"}