{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:XVXQ4IM63EAC4WG77ICCCM3SII","short_pith_number":"pith:XVXQ4IM6","canonical_record":{"source":{"id":"1708.01515","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2017-08-02T20:57:49Z","cross_cats_sorted":[],"title_canon_sha256":"a0e15949aa6fb4efcc7e11d906bedddc2d6db272e02fea73e604c23bd3619ebf","abstract_canon_sha256":"1cd0402f273ee95edd5cadf36bd16d1cfce5c988741874c31c884c3bc00e8f7b"},"schema_version":"1.0"},"canonical_sha256":"bd6f0e219ed9002e58dffa042133724222ae088c20def97fba6c8d433003f7b5","source":{"kind":"arxiv","id":"1708.01515","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1708.01515","created_at":"2026-05-18T00:38:37Z"},{"alias_kind":"arxiv_version","alias_value":"1708.01515v1","created_at":"2026-05-18T00:38:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1708.01515","created_at":"2026-05-18T00:38:37Z"},{"alias_kind":"pith_short_12","alias_value":"XVXQ4IM63EAC","created_at":"2026-05-18T12:31:56Z"},{"alias_kind":"pith_short_16","alias_value":"XVXQ4IM63EAC4WG7","created_at":"2026-05-18T12:31:56Z"},{"alias_kind":"pith_short_8","alias_value":"XVXQ4IM6","created_at":"2026-05-18T12:31:56Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:XVXQ4IM63EAC4WG77ICCCM3SII","target":"record","payload":{"canonical_record":{"source":{"id":"1708.01515","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2017-08-02T20:57:49Z","cross_cats_sorted":[],"title_canon_sha256":"a0e15949aa6fb4efcc7e11d906bedddc2d6db272e02fea73e604c23bd3619ebf","abstract_canon_sha256":"1cd0402f273ee95edd5cadf36bd16d1cfce5c988741874c31c884c3bc00e8f7b"},"schema_version":"1.0"},"canonical_sha256":"bd6f0e219ed9002e58dffa042133724222ae088c20def97fba6c8d433003f7b5","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:38:37.180554Z","signature_b64":"eJjq8Kf5CZqVO/1qqRkECbtVo7iueslq8SIx2dtEfMCKaWEDv7JobKz5gOZD896NRjUXcYtpyXvHEPmaRSJxAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bd6f0e219ed9002e58dffa042133724222ae088c20def97fba6c8d433003f7b5","last_reissued_at":"2026-05-18T00:38:37.179991Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:38:37.179991Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1708.01515","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:38:37Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8xAQZsIyOWHY2ixH98ljNg79ZSN27EHTCLQ52a3Kw8XyL/fbTfY2VLTCR2cEdB4hVZf6KT8Le8yVkGvaYzmnCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T21:47:23.818399Z"},"content_sha256":"cdc34d750977b83b3e52d4fd879a448649e4bb07a96bd62ecb0e1056ed98d33f","schema_version":"1.0","event_id":"sha256:cdc34d750977b83b3e52d4fd879a448649e4bb07a96bd62ecb0e1056ed98d33f"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:XVXQ4IM63EAC4WG77ICCCM3SII","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Cramer's rules for the solution to the two-sided restricted quaternion matrix equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Ivan Kyrchei","submitted_at":"2017-08-02T20:57:49Z","abstract_excerpt":"Weighted singular value decomposition (WSVD) of a quaternion matrix and with its help determinantal representations of the quaternion weighted Moore-Penrose inverse have been derived recently by the author. In this paper, using these determinantal representations, explicit determinantal representation formulas for the solution of the restricted quaternion matrix equations, ${\\bf A}{\\bf X}{\\bf B}={\\bf D}$, and consequently, ${\\bf A}{\\bf X}={\\bf D}$ and ${\\bf X}{\\bf B}={\\bf D}$ are obtained within the framework of the theory of column-row determinants. We consider all possible cases depending on"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1708.01515","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:38:37Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"OpZhgysx7xDUbvVFaQze3p51TTQB0ITW+/WVvrVM72uxQdJMk0mdHnH6MsaZLWXb9bMkApMAKTMNw5rj1TrNCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T21:47:23.818735Z"},"content_sha256":"acc2fb04869a8a0d770dbfd0124d364d218a88a9248e17097256783b66595744","schema_version":"1.0","event_id":"sha256:acc2fb04869a8a0d770dbfd0124d364d218a88a9248e17097256783b66595744"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/XVXQ4IM63EAC4WG77ICCCM3SII/bundle.json","state_url":"https://pith.science/pith/XVXQ4IM63EAC4WG77ICCCM3SII/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/XVXQ4IM63EAC4WG77ICCCM3SII/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-10T21:47:23Z","links":{"resolver":"https://pith.science/pith/XVXQ4IM63EAC4WG77ICCCM3SII","bundle":"https://pith.science/pith/XVXQ4IM63EAC4WG77ICCCM3SII/bundle.json","state":"https://pith.science/pith/XVXQ4IM63EAC4WG77ICCCM3SII/state.json","well_known_bundle":"https://pith.science/.well-known/pith/XVXQ4IM63EAC4WG77ICCCM3SII/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:XVXQ4IM63EAC4WG77ICCCM3SII","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":"1cd0402f273ee95edd5cadf36bd16d1cfce5c988741874c31c884c3bc00e8f7b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2017-08-02T20:57:49Z","title_canon_sha256":"a0e15949aa6fb4efcc7e11d906bedddc2d6db272e02fea73e604c23bd3619ebf"},"schema_version":"1.0","source":{"id":"1708.01515","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1708.01515","created_at":"2026-05-18T00:38:37Z"},{"alias_kind":"arxiv_version","alias_value":"1708.01515v1","created_at":"2026-05-18T00:38:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1708.01515","created_at":"2026-05-18T00:38:37Z"},{"alias_kind":"pith_short_12","alias_value":"XVXQ4IM63EAC","created_at":"2026-05-18T12:31:56Z"},{"alias_kind":"pith_short_16","alias_value":"XVXQ4IM63EAC4WG7","created_at":"2026-05-18T12:31:56Z"},{"alias_kind":"pith_short_8","alias_value":"XVXQ4IM6","created_at":"2026-05-18T12:31:56Z"}],"graph_snapshots":[{"event_id":"sha256:acc2fb04869a8a0d770dbfd0124d364d218a88a9248e17097256783b66595744","target":"graph","created_at":"2026-05-18T00:38:37Z","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":"Weighted singular value decomposition (WSVD) of a quaternion matrix and with its help determinantal representations of the quaternion weighted Moore-Penrose inverse have been derived recently by the author. In this paper, using these determinantal representations, explicit determinantal representation formulas for the solution of the restricted quaternion matrix equations, ${\\bf A}{\\bf X}{\\bf B}={\\bf D}$, and consequently, ${\\bf A}{\\bf X}={\\bf D}$ and ${\\bf X}{\\bf B}={\\bf D}$ are obtained within the framework of the theory of column-row determinants. We consider all possible cases depending on","authors_text":"Ivan Kyrchei","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2017-08-02T20:57:49Z","title":"Cramer's rules for the solution to the two-sided restricted quaternion matrix equation"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1708.01515","kind":"arxiv","version":1},"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:cdc34d750977b83b3e52d4fd879a448649e4bb07a96bd62ecb0e1056ed98d33f","target":"record","created_at":"2026-05-18T00:38:37Z","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":"1cd0402f273ee95edd5cadf36bd16d1cfce5c988741874c31c884c3bc00e8f7b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2017-08-02T20:57:49Z","title_canon_sha256":"a0e15949aa6fb4efcc7e11d906bedddc2d6db272e02fea73e604c23bd3619ebf"},"schema_version":"1.0","source":{"id":"1708.01515","kind":"arxiv","version":1}},"canonical_sha256":"bd6f0e219ed9002e58dffa042133724222ae088c20def97fba6c8d433003f7b5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"bd6f0e219ed9002e58dffa042133724222ae088c20def97fba6c8d433003f7b5","first_computed_at":"2026-05-18T00:38:37.179991Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:38:37.179991Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"eJjq8Kf5CZqVO/1qqRkECbtVo7iueslq8SIx2dtEfMCKaWEDv7JobKz5gOZD896NRjUXcYtpyXvHEPmaRSJxAg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:38:37.180554Z","signed_message":"canonical_sha256_bytes"},"source_id":"1708.01515","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:cdc34d750977b83b3e52d4fd879a448649e4bb07a96bd62ecb0e1056ed98d33f","sha256:acc2fb04869a8a0d770dbfd0124d364d218a88a9248e17097256783b66595744"],"state_sha256":"f493c1422bab72b266b6947387d4b9f356232a5da63a4cb1149bf64c235f79d1"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Rx3S/62TILHpkNKScdi5qWFLSE9g+CHihoaGC7OQTaeH3AQySrK2/s7EF8DlGaSjp+y9qC+/3JMuWHFT4E+iBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-10T21:47:23.820667Z","bundle_sha256":"08b6082947ec1856e0e19327ea5975d16819b467e76599c30d4bd40d0969cb47"}}