{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2023:TLWMXFZG64DGB6HAIPC2GD4ANS","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":"ad6997331783b4bfd2b83a04fe610a207cea1d83d520322b895f04bb427e8daf","cross_cats_sorted":["math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2023-01-31T17:24:46Z","title_canon_sha256":"339b549dbf0fcd2933017c34bca1d1bcc6139bfc34a5a653ff05f097be6b275b"},"schema_version":"1.0","source":{"id":"2301.13776","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2301.13776","created_at":"2026-07-05T06:44:29Z"},{"alias_kind":"arxiv_version","alias_value":"2301.13776v2","created_at":"2026-07-05T06:44:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2301.13776","created_at":"2026-07-05T06:44:29Z"},{"alias_kind":"pith_short_12","alias_value":"TLWMXFZG64DG","created_at":"2026-07-05T06:44:29Z"},{"alias_kind":"pith_short_16","alias_value":"TLWMXFZG64DGB6HA","created_at":"2026-07-05T06:44:29Z"},{"alias_kind":"pith_short_8","alias_value":"TLWMXFZG","created_at":"2026-07-05T06:44:29Z"}],"graph_snapshots":[{"event_id":"sha256:aac2f988832ae06ade3636b95f809671b7bd87a861f119e90a90dccc5eaadc2c","target":"graph","created_at":"2026-07-05T06:44:29Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2301.13776/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Suppose $Q(x)$ is a real $n\\times n$ regular symmetric positive semidefinite matrix polynomial. Then it can be factored as $$Q(x) = G(x)^TG(x),$$ where $G(x)$ is a real $n\\times n$ matrix polynomial with degree half that of $Q(x)$ if and only if $\\det(Q(x))$ is the square of a nonzero real polynomial. We provide a constructive proof of this fact, rooted in finding a skew-symmetric solution to a modified algebraic Riccati equation $$XSX - XR + R^TX + P = 0,$$ where $P,R,S$ are real $n\\times n$ matrices with $P$ and $S$ real symmetric. In addition, we provide a detailed algorithm for computing t","authors_text":"Hugo J. Woerdeman, Sarah Gift","cross_cats":["math.FA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2023-01-31T17:24:46Z","title":"Real Factorization of Positive Semidefinite Matrix Polynomials"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2301.13776","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:d6e0439218382f3e0f3058ed0871ba4ba07b2aeb5310e6fcdc253cb8ff26478d","target":"record","created_at":"2026-07-05T06:44:29Z","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":"ad6997331783b4bfd2b83a04fe610a207cea1d83d520322b895f04bb427e8daf","cross_cats_sorted":["math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2023-01-31T17:24:46Z","title_canon_sha256":"339b549dbf0fcd2933017c34bca1d1bcc6139bfc34a5a653ff05f097be6b275b"},"schema_version":"1.0","source":{"id":"2301.13776","kind":"arxiv","version":2}},"canonical_sha256":"9aeccb9726f70660f8e043c5a30f806c94b769ff92c794650ca50ff18c3f84b2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9aeccb9726f70660f8e043c5a30f806c94b769ff92c794650ca50ff18c3f84b2","first_computed_at":"2026-07-05T06:44:29.471408Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T06:44:29.471408Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"df+TPyd8RRPdaRXZ+bcteK1bbio7f2gzWGQSDtaGJFqiVKOypmI2fH+kk5hsOKMjD1j6tyRGX2ZyBmpODse/AQ==","signature_status":"signed_v1","signed_at":"2026-07-05T06:44:29.471813Z","signed_message":"canonical_sha256_bytes"},"source_id":"2301.13776","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d6e0439218382f3e0f3058ed0871ba4ba07b2aeb5310e6fcdc253cb8ff26478d","sha256:aac2f988832ae06ade3636b95f809671b7bd87a861f119e90a90dccc5eaadc2c"],"state_sha256":"b09335fb447bf54049162bf4b2521831caafc7e21ee06ed99b140fcba8f15f1b"}