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But that third randomized pseudoinverse $A^+_r$ may be very useful when $A$ is a very large matrix.\n  1. $A^+ = R^+C^+$ when $A = CR$ and $C$ has independent columns and $R$ has independent rows.\n  2. $A^+ = (C^+CR)^+(CRR^+)^+$ is always correct.\n  3. $A^+_r = (P^TCR)^+P^TCRQ(CRQ)^+ = A^+$ only when $\\mathrm{rank}(P^TA) = \\mathrm{rank}(AQ) = \\mathrm{rank}(A)$ with $A = CR$."},"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":"2305.01716","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2023-05-02T18:28:22Z","cross_cats_sorted":["cs.NA"],"title_canon_sha256":"90cfc1aab175390433657f2e1a036bda4994964f0ea7c213091e6a516b2f0074","abstract_canon_sha256":"36305736095380350418e16c94e8e4387fb93a5685f5c133fa1435653d656e59"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T08:00:59.258181Z","signature_b64":"ZA4ONP4agcbvUvMdk7Wtf2M+rH1FGU1S2ZVoOVeWtc55yNBHmAb3TGhmP+FozskLU1FMwnB9yYjzMOGit0KQAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9ad55d7ceee694c2f2de266983fdca47a8b5afb7a30a50509e054259596028fe","last_reissued_at":"2026-07-05T08:00:59.257699Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T08:00:59.257699Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Pseudoinverse of $A=CR$ is $A^+=R^+C^+$ (?)","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.NA"],"primary_cat":"math.NA","authors_text":"Gilbert Strang, Micha{\\l} P. 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But that third randomized pseudoinverse $A^+_r$ may be very useful when $A$ is a very large matrix.\n  1. $A^+ = R^+C^+$ when $A = CR$ and $C$ has independent columns and $R$ has independent rows.\n  2. $A^+ = (C^+CR)^+(CRR^+)^+$ is always correct.\n  3. $A^+_r = (P^TCR)^+P^TCRQ(CRQ)^+ = A^+$ only when $\\mathrm{rank}(P^TA) = \\mathrm{rank}(AQ) = \\mathrm{rank}(A)$ with $A = CR$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2305.01716","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2305.01716/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2305.01716","created_at":"2026-07-05T08:00:59.257754+00:00"},{"alias_kind":"arxiv_version","alias_value":"2305.01716v3","created_at":"2026-07-05T08:00:59.257754+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2305.01716","created_at":"2026-07-05T08:00:59.257754+00:00"},{"alias_kind":"pith_short_12","alias_value":"TLKV27HO42KM","created_at":"2026-07-05T08:00:59.257754+00:00"},{"alias_kind":"pith_short_16","alias_value":"TLKV27HO42KMF4W6","created_at":"2026-07-05T08:00:59.257754+00:00"},{"alias_kind":"pith_short_8","alias_value":"TLKV27HO","created_at":"2026-07-05T08:00:59.257754+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2403.01919","citing_title":"Randomized Approach to Matrix Completion: Applications in Recommendation Systems and Image Inpainting","ref_index":35,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/TLKV27HO42KMF4W6EZUYH7OKI6","json":"https://pith.science/pith/TLKV27HO42KMF4W6EZUYH7OKI6.json","graph_json":"https://pith.science/api/pith-number/TLKV27HO42KMF4W6EZUYH7OKI6/graph.json","events_json":"https://pith.science/api/pith-number/TLKV27HO42KMF4W6EZUYH7OKI6/events.json","paper":"https://pith.science/paper/TLKV27HO"},"agent_actions":{"view_html":"https://pith.science/pith/TLKV27HO42KMF4W6EZUYH7OKI6","download_json":"https://pith.science/pith/TLKV27HO42KMF4W6EZUYH7OKI6.json","view_paper":"https://pith.science/paper/TLKV27HO","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2305.01716&json=true","fetch_graph":"https://pith.science/api/pith-number/TLKV27HO42KMF4W6EZUYH7OKI6/graph.json","fetch_events":"https://pith.science/api/pith-number/TLKV27HO42KMF4W6EZUYH7OKI6/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/TLKV27HO42KMF4W6EZUYH7OKI6/action/timestamp_anchor","attest_storage":"https://pith.science/pith/TLKV27HO42KMF4W6EZUYH7OKI6/action/storage_attestation","attest_author":"https://pith.science/pith/TLKV27HO42KMF4W6EZUYH7OKI6/action/author_attestation","sign_citation":"https://pith.science/pith/TLKV27HO42KMF4W6EZUYH7OKI6/action/citation_signature","submit_replication":"https://pith.science/pith/TLKV27HO42KMF4W6EZUYH7OKI6/action/replication_record"}},"created_at":"2026-07-05T08:00:59.257754+00:00","updated_at":"2026-07-05T08:00:59.257754+00:00"}