{ "schema_version": 1, "kind": "ConceptView", "producer": "concept-view-1", "concept_id": "urn:uuid:60808942-4f7a-44c9-85de-4b3cf47ec7dd", "labels": { "zh-CN": "正定", "en": "positive definite" }, "alternative_labels": [], "scope_note": "", "identity_status": "active", "identity_revision": "6f0c1cdc779a5fe980bda4e5a328b17d592ffd64659203788ece7211b688aff1", "occurrences": [ { "concept_id": "urn:uuid:60808942-4f7a-44c9-85de-4b3cf47ec7dd", "book_id": "szekeres", "edition_id": "source-snapshot-bc3404156f6e", "lang": "zh-CN", "route": "zh/chapter-05/01-real-inner-product-spaces", "anchor": "concept-5-positive-definite", "chapter": "5", "section": "5.1", "offset": 2271, "range": [ 7760, 7998 ], "source_revision": "b0905ca500407f5be9e556fa6537e88d1d774fe93ee3047ac3c1ce4bc6352512", "source_category": "translation", "review_status": "unreviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "正定(positive definite)", "context": { "range": [ 7753, 8396 ], "quote": "此时称该内积为正定(positive definite)或欧氏内积，而定义了这种内积的向量空间$V$称为欧氏向量空间(Euclidean vector space)。条件(RIP3')蕴含条件(RIP3)，因为若存在非零向量$u$使得对所有$v \\in V$都有$u \\cdot v = 0$，则$u \\cdot u = 0$（取$v = u )$，这违反(RIP3')）。因此正定性是比非奇异性更强的要求。", "quote_sha256": "2507da321e8b0f61b134693f3da072222bc84f7f9b868f0d42e61365e7228b5a", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "en/chapter-05/01-real-inner-product-spaces", "lang": "en", "source_category": "original_transcription", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-76ba2e2a7e7e72d7daa34f25" }, { "concept_id": "urn:uuid:60808942-4f7a-44c9-85de-4b3cf47ec7dd", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "en", "route": "books/woit-qtgr/2025-10-20/en/chapter-04/04-inner-products", "anchor": "first-128346b487443b107031-en", "chapter": "4", "section": "4.4", "offset": 1637, "range": [ 10515, 10828 ], "source_revision": "3977807e255d2714286f744389ed0a4b92c7ca420946f9f415e6e63c1b284cdf", "source_category": "original_transcription", "review_status": "ai-transcription-compared", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "positive definite", "context": { "range": [ 10407, 11001 ], "quote": "Our real inner products will usually be [positive definite] ($(v,v)\\geq 0$ and $(v,v)=0$ if and only if $v=0$). 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