{ "schema_version": 1, "kind": "ConceptView", "producer": "concept-view-1", "concept_id": "urn:uuid:fc0ae0df-4043-4beb-be49-1664bb2bfb3c", "labels": { "zh-CN": "西尔维斯特惯性定律", "en": "Sylvester’s law of inertia" }, "alternative_labels": [], "scope_note": "", "identity_status": "active", "identity_revision": "8a08719de2898f533cfe9edf2dfd9681c38e1866d1c4e419253002e126bddc6b", "occurrences": [ { "concept_id": "urn:uuid:fc0ae0df-4043-4beb-be49-1664bb2bfb3c", "book_id": "szekeres", "edition_id": "source-snapshot-bc3404156f6e", "lang": "zh-CN", "route": "zh/chapter-05/01-real-inner-product-spaces", "anchor": "first-777e78796e7eacd431b7-zh", "chapter": "5", "section": "5.1", "offset": 12652, "range": [ 18141, 19348 ], "source_revision": "b0905ca500407f5be9e556fa6537e88d1d774fe93ee3047ac3c1ce4bc6352512", "source_category": "translation", "review_status": "unreviewed", "role": "theorem", "role_basis": { "kind": "registered-source-environment", "registered_position": { "route": "zh/chapter-05/01-real-inner-product-spaces", "lang": "zh-CN", "section": "5.1", "signal": "named-theorem", "label": "西尔维斯特惯性定律", "english": "Sylvester’s law of inertia", "offset": 10412 } }, "label": "> [!theorem] 定理 5.3 · 西尔维斯特惯性定律 (Sylvester’s law of inertia)", "context": { "range": [ 18141, 19348 ], "quote": "\n\n> [!theorem] 定理 5.3 · 西尔维斯特惯性定律 (Sylvester’s law of inertia)\n>\n> (西尔维斯特) 在 $\\eta _ { i }$ 中，$+$ 与 $-$ 符号的数目与标准正交基的选取无关。\n\n> [!proof]- 证明\n>\n> 设 $\\{ e _ { i } \\}$ 与 $\\{ f _ { j } \\}$ 为两个标准正交基，使得\n>\n> $$\n> e _ {1} \\cdot e _ {1} = \\dots = e _ {r} \\cdot e _ {r} = + 1, \\qquad e _ {r + 1} \\cdot e _ {r + 1} = \\dots = e _ {n} \\cdot e _ {n} = - 1,\n> $$\n>\n> $$\n> f _ {1} \\cdot f _ {1} = \\dots = f _ {s} \\cdot f _ {s} = + 1, \\quad f _ {s + 1} \\cdot f _ {s + 1} = \\dots = f _ {n} \\cdot f _ {n} = - 1.\n> $$\n>\n> 若 $s > r$，则向量 $f _ { 1 } , \\ldots , f _ { s }$ 与 $e _ { r + 1 } , \\ldots , e _ { n }$ 构成一组 $s + n - r > n = \\dim V$ 个向量，且它们之间必存在一个非平凡线性关系，\n>\n> $$\n> a ^ {1} f _ {1} + \\dots + a ^ {s} f _ {s} + b ^ {1} e _ {r + 1} + \\dots + b ^ {n - r} e _ {n} = 0.\n> $$\n>\n> 由于$a ^ { i }$不可能全部为零，因为$e _ { i }$构成线性无关集。类似地，并非所有$b ^ { j }$都会为零。令\n>\n> $$\n> u = a ^ {1} f _ {1} + \\dots + a ^ {s} f _ {s} = - b ^ {1} e _ {r + 1} - \\dots - b ^ {n - r} e _ {n} \\neq 0\n> $$\n>\n> 我们得到了矛盾\n>\n> $$\n> u \\cdot u = \\sum_ {i = 1} ^ {s} (a ^ {i}) ^ {2} > 0 \\quad \\text { and } \\quad u \\cdot u = - \\sum_ {j = 1} ^ {n - r} (b ^ {j}) ^ {2} < 0.\n> $$\n>\n> 因此，$r = s$ 与这两个基必须具有完全相同数量的 $+$ 和 $-$ 符号。\n", "quote_sha256": "06e28c96e0600eae51eafd234960ea8c358697fa76a147ad7036ede3c88255f6", "environment": "theorem", "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-269603c1d0a696e183a60fb8" }, { "concept_id": "urn:uuid:fc0ae0df-4043-4beb-be49-1664bb2bfb3c", "book_id": "szekeres", "edition_id": "source-snapshot-bc3404156f6e", "lang": "en", "route": "en/chapter-05/01-real-inner-product-spaces", "anchor": "first-777e78796e7eacd431b7-en", "chapter": "5", "section": "5.1", "offset": 14149, "range": [ 19131, 20623 ], "source_revision": "468c3cd8debdde609d01ed3d0c5b4ba8e0b32580c271b042cf7fbba0fc095d05", "source_category": "original_transcription", "review_status": "unreviewed", "role": "theorem", "role_basis": { "kind": "registered-source-environment", "registered_position": { "route": "en/chapter-05/01-real-inner-product-spaces", "lang": "en", "section": "5.1", "signal": "named-theorem", "label": "Sylvester’s law of inertia", "english": "Sylvester’s law of inertia", "offset": 13937 } }, "label": "> [!theorem] Theorem 5.3 · Sylvester’s law of inertia", "context": { "range": [ 19131, 20623 ], "quote": "\n\n> [!theorem] Theorem 5.3 · Sylvester’s law of inertia\n>\n> (Sylvester) The number of $+$ and $-$ signs among the $\\eta _ { i }$ is independent of the choice of orthonormal basis.\n\n> [!proof]- Proof\n>\n> Let $\\{ e _ { i } \\}$ and $\\{ f _ { j } \\}$ be two orthonormal bases such that\n>\n> $$\n> e _ {1} \\cdot e _ {1} = \\dots = e _ {r} \\cdot e _ {r} = + 1, \\qquad e _ {r + 1} \\cdot e _ {r + 1} = \\dots = e _ {n} \\cdot e _ {n} = - 1,\n> $$\n>\n> $$\n> f _ {1} \\cdot f _ {1} = \\dots = f _ {s} \\cdot f _ {s} = + 1, \\quad f _ {s + 1} \\cdot f _ {s + 1} = \\dots = f _ {n} \\cdot f _ {n} = - 1.\n> $$\n>\n> If $s > r$ then the vectors $f _ { 1 } , \\ldots , f _ { s }$ and $e _ { r + 1 } , \\ldots , e _ { n }$ are a set of $s + n - r > n = \\dim V$ vectors and there must be a non-trivial linear relation between them,\n>\n> $$\n> a ^ {1} f _ {1} + \\dots + a ^ {s} f _ {s} + b ^ {1} e _ {r + 1} + \\dots + b ^ {n - r} e _ {n} = 0.\n> $$\n>\n> The $a ^ { i }$ cannot all vanish since the $e _ { i }$ form an l.i. set. Similarly, not all the $b ^ { j }$ will vanish. Setting\n>\n> $$\n> u = a ^ {1} f _ {1} + \\dots + a ^ {s} f _ {s} = - b ^ {1} e _ {r + 1} - \\dots - b ^ {n - r} e _ {n} \\neq 0\n> $$\n>\n> we have the contradiction\n>\n> $$\n> u \\cdot u = \\sum_ {i = 1} ^ {s} (a ^ {i}) ^ {2} > 0 \\quad \\text { and } \\quad u \\cdot u = - \\sum_ {j = 1} ^ {n - r} (b ^ {j}) ^ {2} < 0.\n> $$\n>\n> Hence $r = s$ and the two bases must have exactly the same number of $+$ and $-$ signs.\n", "quote_sha256": "c16e3267929b5814f1543463f4c897acec5313e2b48cd935a7bad7dc20eb8815", "environment": "theorem", "excerpt_omitted": false }, "parallel_readings": [ { "route": "zh/chapter-05/01-real-inner-product-spaces", "lang": "zh-CN", "source_category": "translation", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-bae393c046724c5ed1c96880" } ], "registered_positions": [ { "route": "en/chapter-05/01-real-inner-product-spaces", "lang": "en", "section": "5.1", "signal": "named-theorem", "label": "Sylvester’s law of inertia", "english": "Sylvester’s law of inertia", "offset": 13937 }, { "route": "zh/chapter-05/01-real-inner-product-spaces", "lang": "zh-CN", "section": "5.1", "signal": "named-theorem", "label": "西尔维斯特惯性定律", "english": "Sylvester’s law of inertia", "offset": 10412 } ], "statements": [], "resolved_references": [ { "raw": "href=\"#theorem-5-3\"", "href": "#theorem-5-3", "range": [ 18184, 18203 ], "relation": "references", "target_route": "zh/chapter-05/01-real-inner-product-spaces", "target_anchor": "theorem-5-3", "resolved": true, "reason": null, "source_revision": "b0905ca500407f5be9e556fa6537e88d1d774fe93ee3047ac3c1ce4bc6352512", "source_route": "zh/chapter-05/01-real-inner-product-spaces", "occurrence_id": "occ-269603c1d0a696e183a60fb8" }, { "raw": "href=\"#theorem-5-3\"", "href": "#theorem-5-3", "range": [ 19174, 19193 ], "relation": "references", "target_route": "en/chapter-05/01-real-inner-product-spaces", "target_anchor": "theorem-5-3", "resolved": true, "reason": null, "source_revision": "468c3cd8debdde609d01ed3d0c5b4ba8e0b32580c271b042cf7fbba0fc095d05", "source_route": "en/chapter-05/01-real-inner-product-spaces", "occurrence_id": "occ-bae393c046724c5ed1c96880" } ], "reading_order": [ "occ-269603c1d0a696e183a60fb8", "occ-bae393c046724c5ed1c96880" ], "recorded_dependencies": [], "reading_paths": [], "coverage": { "registered_positions": 2, "located_occurrences": 2, "whole_book_search_complete": false, "registered_positions_truncated": false, "independent_human_review_implied": false }, "unresolved": [], "base_revision": "7e2931c91e454a85254f8b1beb127192403b14bf99650499c77fa3527882d400", "optional_enrichments": [] }