{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:GO663DLRYY44Y7SCQZHZ5UXXYE","short_pith_number":"pith:GO663DLR","schema_version":"1.0","canonical_sha256":"33bded8d71c639cc7e42864f9ed2f7c106348bd91c17623c54b1d02c3133c8a8","source":{"kind":"arxiv","id":"1608.04491","version":1},"attestation_state":"computed","paper":{"title":"Computing the Fr\\'echet Derivative of the Polar Decomposition","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Evan S. Gawlik, Melvin Leok","submitted_at":"2016-08-16T05:59:29Z","abstract_excerpt":"We derive iterative methods for computing the Fr\\'{e}chet derivative of the map which sends a full-rank matrix $A$ to the factor $U$ in its polar decomposition $A=UH$, where $U$ has orthonormal columns and $H$ is Hermitian positive definite. The methods apply to square matrices as well as rectangular matrices having more rows than columns. Our derivation relies on a novel identity that relates the Fr\\'{e}chet derivative of the polar decomposition to the matrix sign function $\\mathrm{sign}(X) = X (X^2)^{-1/2}$ applied to a certain block matrix $X$."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1608.04491","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2016-08-16T05:59:29Z","cross_cats_sorted":[],"title_canon_sha256":"2720a74cb4f5c8a40f5ee1454f12e2365c655193a5b7877bcba8eb967df82ebe","abstract_canon_sha256":"4653644923acb24e22924836e03027e8d6a9bb96c10c54ad16d745fe296a79c1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:08:37.088800Z","signature_b64":"rryLhJKCw+Jz80hkaGYTVATDSt7FVb2dEHKUgqEzXQ3Zmj1PAvScXehiLk0Ls+WJ5LGsv8/J6VF1fzZHyUqQAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"33bded8d71c639cc7e42864f9ed2f7c106348bd91c17623c54b1d02c3133c8a8","last_reissued_at":"2026-05-18T01:08:37.088097Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:08:37.088097Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Computing the Fr\\'echet Derivative of the Polar Decomposition","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Evan S. Gawlik, Melvin Leok","submitted_at":"2016-08-16T05:59:29Z","abstract_excerpt":"We derive iterative methods for computing the Fr\\'{e}chet derivative of the map which sends a full-rank matrix $A$ to the factor $U$ in its polar decomposition $A=UH$, where $U$ has orthonormal columns and $H$ is Hermitian positive definite. The methods apply to square matrices as well as rectangular matrices having more rows than columns. Our derivation relies on a novel identity that relates the Fr\\'{e}chet derivative of the polar decomposition to the matrix sign function $\\mathrm{sign}(X) = X (X^2)^{-1/2}$ applied to a certain block matrix $X$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.04491","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1608.04491","created_at":"2026-05-18T01:08:37.088199+00:00"},{"alias_kind":"arxiv_version","alias_value":"1608.04491v1","created_at":"2026-05-18T01:08:37.088199+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1608.04491","created_at":"2026-05-18T01:08:37.088199+00:00"},{"alias_kind":"pith_short_12","alias_value":"GO663DLRYY44","created_at":"2026-05-18T12:30:19.053100+00:00"},{"alias_kind":"pith_short_16","alias_value":"GO663DLRYY44Y7SC","created_at":"2026-05-18T12:30:19.053100+00:00"},{"alias_kind":"pith_short_8","alias_value":"GO663DLR","created_at":"2026-05-18T12:30:19.053100+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/GO663DLRYY44Y7SCQZHZ5UXXYE","json":"https://pith.science/pith/GO663DLRYY44Y7SCQZHZ5UXXYE.json","graph_json":"https://pith.science/api/pith-number/GO663DLRYY44Y7SCQZHZ5UXXYE/graph.json","events_json":"https://pith.science/api/pith-number/GO663DLRYY44Y7SCQZHZ5UXXYE/events.json","paper":"https://pith.science/paper/GO663DLR"},"agent_actions":{"view_html":"https://pith.science/pith/GO663DLRYY44Y7SCQZHZ5UXXYE","download_json":"https://pith.science/pith/GO663DLRYY44Y7SCQZHZ5UXXYE.json","view_paper":"https://pith.science/paper/GO663DLR","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1608.04491&json=true","fetch_graph":"https://pith.science/api/pith-number/GO663DLRYY44Y7SCQZHZ5UXXYE/graph.json","fetch_events":"https://pith.science/api/pith-number/GO663DLRYY44Y7SCQZHZ5UXXYE/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/GO663DLRYY44Y7SCQZHZ5UXXYE/action/timestamp_anchor","attest_storage":"https://pith.science/pith/GO663DLRYY44Y7SCQZHZ5UXXYE/action/storage_attestation","attest_author":"https://pith.science/pith/GO663DLRYY44Y7SCQZHZ5UXXYE/action/author_attestation","sign_citation":"https://pith.science/pith/GO663DLRYY44Y7SCQZHZ5UXXYE/action/citation_signature","submit_replication":"https://pith.science/pith/GO663DLRYY44Y7SCQZHZ5UXXYE/action/replication_record"}},"created_at":"2026-05-18T01:08:37.088199+00:00","updated_at":"2026-05-18T01:08:37.088199+00:00"}