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For  such matrices to exist, one must have $<v_k, v_l> \\geq 0$ for all $k,l=1,..., n$.  We prove that if $n<5$ then this trivial necessary condition is also a sufficient one and find an appropriate example showing that from $n=5$ this is not so --- even if we allowed realizations by positive operators in a von Neumann algebra with a faithful normal traci"},"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":"1004.0686","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2010-04-05T19:32:40Z","cross_cats_sorted":[],"title_canon_sha256":"cc5a0be7cd18514114e3b1ca35759162209b9d36dbb65fb4b00ac5ac451ad2cf","abstract_canon_sha256":"3e4fca88dd0dbf0f7d5482807d9bc6d38112f1388e31a838e68cca96d2c0a96c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:45:41.379229Z","signature_b64":"12P+3eCAZFGOqiLAHojhWfjre2cjWSnlOiXP+Yp9XJSINmbbcC8S3C/yOi4tmDShBHIeFO07d3GHodrWTWQECA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"eb46a4f652bcba9793eeff55165025e11e5613ecbc1ae75dcb576f1a01e4f8c4","last_reissued_at":"2026-05-18T02:45:41.378829Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:45:41.378829Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On vector configurations that can be realized in the cone of positive matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OA","authors_text":"Mih\\'aly Weiner, P\\'eter E. Frenkel","submitted_at":"2010-04-05T19:32:40Z","abstract_excerpt":"Let $v_1$,..., $v_n$ be $n$ vectors in an inner product space. Can we find a natural number $d$ and positive (semidefinite) complex matrices $A_1$,..., $A_n$ of size $d \\times d$ such that ${\\rm Tr}(A_kA_l)= <v_k, v_l>$ for all $k,l=1,..., n$? For  such matrices to exist, one must have $<v_k, v_l> \\geq 0$ for all $k,l=1,..., n$.  We prove that if $n<5$ then this trivial necessary condition is also a sufficient one and find an appropriate example showing that from $n=5$ this is not so --- even if we allowed realizations by positive operators in a von Neumann algebra with a faithful normal traci"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1004.0686","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":"1004.0686","created_at":"2026-05-18T02:45:41.378886+00:00"},{"alias_kind":"arxiv_version","alias_value":"1004.0686v1","created_at":"2026-05-18T02:45:41.378886+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1004.0686","created_at":"2026-05-18T02:45:41.378886+00:00"},{"alias_kind":"pith_short_12","alias_value":"5NDKJ5SSXS5J","created_at":"2026-05-18T12:26:04.259169+00:00"},{"alias_kind":"pith_short_16","alias_value":"5NDKJ5SSXS5JPE7O","created_at":"2026-05-18T12:26:04.259169+00:00"},{"alias_kind":"pith_short_8","alias_value":"5NDKJ5SS","created_at":"2026-05-18T12:26:04.259169+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/5NDKJ5SSXS5JPE7O75KRMUBF4E","json":"https://pith.science/pith/5NDKJ5SSXS5JPE7O75KRMUBF4E.json","graph_json":"https://pith.science/api/pith-number/5NDKJ5SSXS5JPE7O75KRMUBF4E/graph.json","events_json":"https://pith.science/api/pith-number/5NDKJ5SSXS5JPE7O75KRMUBF4E/events.json","paper":"https://pith.science/paper/5NDKJ5SS"},"agent_actions":{"view_html":"https://pith.science/pith/5NDKJ5SSXS5JPE7O75KRMUBF4E","download_json":"https://pith.science/pith/5NDKJ5SSXS5JPE7O75KRMUBF4E.json","view_paper":"https://pith.science/paper/5NDKJ5SS","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1004.0686&json=true","fetch_graph":"https://pith.science/api/pith-number/5NDKJ5SSXS5JPE7O75KRMUBF4E/graph.json","fetch_events":"https://pith.science/api/pith-number/5NDKJ5SSXS5JPE7O75KRMUBF4E/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/5NDKJ5SSXS5JPE7O75KRMUBF4E/action/timestamp_anchor","attest_storage":"https://pith.science/pith/5NDKJ5SSXS5JPE7O75KRMUBF4E/action/storage_attestation","attest_author":"https://pith.science/pith/5NDKJ5SSXS5JPE7O75KRMUBF4E/action/author_attestation","sign_citation":"https://pith.science/pith/5NDKJ5SSXS5JPE7O75KRMUBF4E/action/citation_signature","submit_replication":"https://pith.science/pith/5NDKJ5SSXS5JPE7O75KRMUBF4E/action/replication_record"}},"created_at":"2026-05-18T02:45:41.378886+00:00","updated_at":"2026-05-18T02:45:41.378886+00:00"}