{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:T3OXVSZRXHI36HN4IEC3M7UVVA","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"e08bb116936e27b1126c80cfe7cb6d7c6ad1faa358beaee15a839958cb1d63a0","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2013-10-30T12:42:40Z","title_canon_sha256":"1d0492a63deef55b131d6d90989a260d2beb3b9a6441df7984cb859e2da983f6"},"schema_version":"1.0","source":{"id":"1310.8124","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1310.8124","created_at":"2026-05-18T03:08:24Z"},{"alias_kind":"arxiv_version","alias_value":"1310.8124v2","created_at":"2026-05-18T03:08:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1310.8124","created_at":"2026-05-18T03:08:24Z"},{"alias_kind":"pith_short_12","alias_value":"T3OXVSZRXHI3","created_at":"2026-05-18T12:27:59Z"},{"alias_kind":"pith_short_16","alias_value":"T3OXVSZRXHI36HN4","created_at":"2026-05-18T12:27:59Z"},{"alias_kind":"pith_short_8","alias_value":"T3OXVSZR","created_at":"2026-05-18T12:27:59Z"}],"graph_snapshots":[{"event_id":"sha256:8189a7b04791bfc1ac5e51b7944d92a036488db38fc78577b27938e32d54a93f","target":"graph","created_at":"2026-05-18T03:08:24Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"In this paper, we derive a formula to compute the solution of the linear matrix equation $X=Af(X)B+C$ via finding any solution of a specific Stein matrix equation $\\mathcal{X}=\\mathcal{A} \\mathcal{X} \\mathcal{B}+\\mathcal{C}$, where the linear (or anti-linear) matrix operator $f$ is period-$n$. According to this formula, we should pay much attention to solve the Stein matrix equation from recently famous numerical methods. For instance, Smith-type iterations, Bartels-Stewart algorithm, and etc.. Moreover, this transformation is used to provide necessary and sufficient conditions of the solvable","authors_text":"Chun-Yueh Chiang","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2013-10-30T12:42:40Z","title":"On the solution of the linear matrix equation $X=Af(X)B+C$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1310.8124","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:33da470b2bee1f83586a12da797cfb667877598431f1046674f6503ab6775ba4","target":"record","created_at":"2026-05-18T03:08:24Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"e08bb116936e27b1126c80cfe7cb6d7c6ad1faa358beaee15a839958cb1d63a0","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2013-10-30T12:42:40Z","title_canon_sha256":"1d0492a63deef55b131d6d90989a260d2beb3b9a6441df7984cb859e2da983f6"},"schema_version":"1.0","source":{"id":"1310.8124","kind":"arxiv","version":2}},"canonical_sha256":"9edd7acb31b9d1bf1dbc4105b67e95a820472259f6d393283f26fd6ef22b4a77","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9edd7acb31b9d1bf1dbc4105b67e95a820472259f6d393283f26fd6ef22b4a77","first_computed_at":"2026-05-18T03:08:24.411434Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:08:24.411434Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"NnqTXCZckouhucwtQpCjRQYhaCCFx6OWEnLEfiOHBAVWdUPsddCh5GlNPfJYL4TBin/hwIrM1q3+26vL+BVQAA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:08:24.411995Z","signed_message":"canonical_sha256_bytes"},"source_id":"1310.8124","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:33da470b2bee1f83586a12da797cfb667877598431f1046674f6503ab6775ba4","sha256:8189a7b04791bfc1ac5e51b7944d92a036488db38fc78577b27938e32d54a93f"],"state_sha256":"7b0ad89b0e0bf4d4bf22e1e5dc315b6d9ca7b96778e26f2163efa2c8b911b2c3"}