{ "schema_version": 1, "kind": "ConceptView", "producer": "concept-view-1", "concept_id": "urn:uuid:1510ce98-1ea2-4c9a-8847-10f4ea3df1e4", "labels": { "zh-CN": "对称算符", "en": "symmetric operator" }, "alternative_labels": [], "scope_note": "", "identity_status": "active", "identity_revision": "a629b0e2a2e8524f017e6cb822f4ff2c3b02ec2fdb6a778669979a1acebd67fb", "occurrences": [ { "concept_id": "urn:uuid:1510ce98-1ea2-4c9a-8847-10f4ea3df1e4", "book_id": "szekeres", "edition_id": "source-snapshot-bc3404156f6e", "lang": "zh-CN", "route": "zh/chapter-05/01-real-inner-product-spaces", "anchor": "concept-5-symmetric-operator", "chapter": "5", "section": "5.1", "offset": 20222, "range": [ 25711, 25953 ], "source_revision": "b0905ca500407f5be9e556fa6537e88d1d774fe93ee3047ac3c1ce4bc6352512", "source_category": "translation", "review_status": "unreviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "对称算符(symmetric operator)", "context": { "range": [ 25630, 26270 ], "quote": "> [!exercise] 习题 5.4 \n>\n> 定义“对称算符(symmetric operator) $S : V \\to V$”这一概念为满足以下条件的算符：\n>\n> $$\n> (S u) \\cdot v = u \\cdot (S v) \\quad \\text { for all } u, v \\in V.\n> $$\n>\n> 证明这导致该分量方程\n>\n> $$\n> S _ {i} ^ {k} g _ {k j} = g _ {i k} S _ {j} ^ {k},\n> $$\n>\n> 等价于矩阵方程\n>\n> $$\n> \\mathsf {S} ^ {T} \\mathsf {G} = \\mathsf {G S}.\n> $$\n>\n> 证明：在欧氏空间中，对于标准正交基，这导致通常的对称性概念，但在伪欧氏空间中则不成立。", "quote_sha256": "a9cdd754bcbff633f3236ea2f215176683b583cc9334444cf1339c90b9e93c12", "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; 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