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We prove the following weak log majorization result: \\begin{equation*} \\lambda (C^{-1}_1D_1\\oplus \\cdots \\oplus C^{-1}_kD_k)\\prec_{w \\,\\log} \\lambda(C^{-1}D), \\end{equation*} where $\\lambda(A)$ denotes the vector of eigenvalues of $A\\in \\Cnn$. The inequality does not hold if one replaces the vectors of eigenvalues"},"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":"1611.05108","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.FA","submitted_at":"2016-11-16T01:04:22Z","cross_cats_sorted":[],"title_canon_sha256":"887b571c33eac108bc6871d7062be00207c30b29f465c750228a6e0118df2c20","abstract_canon_sha256":"a79545942414788e6a315fa6259f8d9f697ad042b5b07cee1fd80b2a06d0fa13"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:58:27.948082Z","signature_b64":"m88zpLwBqkWNEymYYdiWmBbpwEIXVnv6gpz6UtLbeUU60i3VP/e4y5gtxnbVO6Qr/GlhiZjPP5aejV0Y6teiDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ee63349d6d5846ba6a2019647a4b6eb8c6cc66dfb691e44dc761586a8e144d81","last_reissued_at":"2026-05-18T00:58:27.947479Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:58:27.947479Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Weak log majorization and determinantal inequalities","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Pingping Zhang, Tin-Yau Tam","submitted_at":"2016-11-16T01:04:22Z","abstract_excerpt":"Denote by $\\P_n$ the set of $n\\times n$ positive definite matrices. Let $D = D_1\\oplus \\dots \\oplus D_k$, where $D_1\\in \\P_{n_1}, \\dots, D_k \\in \\P_{n_k}$ with $n_1+\\cdots + n_k=n$. Partition $C\\in \\P_n$ according to $(n_1, \\dots, n_k)$ so that $\\Diag C = C_1\\oplus \\dots \\oplus C_k$. We prove the following weak log majorization result: \\begin{equation*} \\lambda (C^{-1}_1D_1\\oplus \\cdots \\oplus C^{-1}_kD_k)\\prec_{w \\,\\log} \\lambda(C^{-1}D), \\end{equation*} where $\\lambda(A)$ denotes the vector of eigenvalues of $A\\in \\Cnn$. The inequality does not hold if one replaces the vectors of eigenvalues"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.05108","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":"1611.05108","created_at":"2026-05-18T00:58:27.947561+00:00"},{"alias_kind":"arxiv_version","alias_value":"1611.05108v1","created_at":"2026-05-18T00:58:27.947561+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1611.05108","created_at":"2026-05-18T00:58:27.947561+00:00"},{"alias_kind":"pith_short_12","alias_value":"5ZRTJHLNLBDL","created_at":"2026-05-18T12:30:01.593930+00:00"},{"alias_kind":"pith_short_16","alias_value":"5ZRTJHLNLBDLU2RA","created_at":"2026-05-18T12:30:01.593930+00:00"},{"alias_kind":"pith_short_8","alias_value":"5ZRTJHLN","created_at":"2026-05-18T12:30:01.593930+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/5ZRTJHLNLBDLU2RADFSHUS3OXD","json":"https://pith.science/pith/5ZRTJHLNLBDLU2RADFSHUS3OXD.json","graph_json":"https://pith.science/api/pith-number/5ZRTJHLNLBDLU2RADFSHUS3OXD/graph.json","events_json":"https://pith.science/api/pith-number/5ZRTJHLNLBDLU2RADFSHUS3OXD/events.json","paper":"https://pith.science/paper/5ZRTJHLN"},"agent_actions":{"view_html":"https://pith.science/pith/5ZRTJHLNLBDLU2RADFSHUS3OXD","download_json":"https://pith.science/pith/5ZRTJHLNLBDLU2RADFSHUS3OXD.json","view_paper":"https://pith.science/paper/5ZRTJHLN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1611.05108&json=true","fetch_graph":"https://pith.science/api/pith-number/5ZRTJHLNLBDLU2RADFSHUS3OXD/graph.json","fetch_events":"https://pith.science/api/pith-number/5ZRTJHLNLBDLU2RADFSHUS3OXD/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/5ZRTJHLNLBDLU2RADFSHUS3OXD/action/timestamp_anchor","attest_storage":"https://pith.science/pith/5ZRTJHLNLBDLU2RADFSHUS3OXD/action/storage_attestation","attest_author":"https://pith.science/pith/5ZRTJHLNLBDLU2RADFSHUS3OXD/action/author_attestation","sign_citation":"https://pith.science/pith/5ZRTJHLNLBDLU2RADFSHUS3OXD/action/citation_signature","submit_replication":"https://pith.science/pith/5ZRTJHLNLBDLU2RADFSHUS3OXD/action/replication_record"}},"created_at":"2026-05-18T00:58:27.947561+00:00","updated_at":"2026-05-18T00:58:27.947561+00:00"}