{ "schema_version": 1, "kind": "ConceptView", "producer": "concept-view-1", "concept_id": "urn:uuid:45d0c13b-cde3-424f-8e70-a8c56a307519", "labels": { "zh-CN": "伴随矩阵", "en": "adjoint matrix" }, "alternative_labels": [ { "lang": "en", "value": "adjoint" } ], "scope_note": "", "identity_status": "active", "identity_revision": "7fb0a3107a06edfc15ba9b856873c36405d2118e6403ac315f5777e3ad335fe1", "occurrences": [ { "concept_id": "urn:uuid:45d0c13b-cde3-424f-8e70-a8c56a307519", "book_id": "szekeres", "edition_id": "source-snapshot-bc3404156f6e", "lang": "zh-CN", "route": "zh/chapter-02/03-matrix-groups", "anchor": "concept-2-adjoint", "chapter": "2", "section": "2.3", "offset": 14428, "range": [ 20749, 20969 ], "source_revision": "3dd2476e51de9c40a8d6ddb8f88871c179362c4e1209a1a91c0b6b3f4e535a63", "source_category": "translation", "review_status": "unreviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "伴随矩阵(adjoint)", "context": { "range": [ 20653, 22192 ], "quote": "> [!example] 例 2.15 \n>\n> 复矩阵 $\\mathsf{A}$ 的伴随矩阵(adjoint)（共轭转置）定义为 $\\mathsf{A}^{\\dagger}=\\overline{\\mathsf{A}^{T}}$，分量为 $a_{ij}^{\\dagger}=\\overline{a_{ji}}$；复数上的横线表示复共轭。若 $n\\times n$ 复矩阵 $\\mathsf{U}$ 满足\n>\n> \n>\n> $$\n> \\mathsf{U}\\mathsf{U}^{\\dagger}=\\mathsf{I},\\tag{2.12}\n> $$\n>\n> 则称它为酉矩阵(unitary matrix)。立即可得\n>\n> $$\n> \\det\\mathsf{U}\\,\\overline{\\det\\mathsf{U}}=|\\det\\mathsf{U}|^{2}=1.\n> $$\n>\n> 因而存在实数 $\\phi\\in[0,2\\pi)$，使 $\\det\\mathsf{U}=e^{i\\phi}$。所有酉矩阵都非奇异，群性质也容易验证。全体 $n\\times n$ 酉矩阵组成$n$ 阶酉群(unitary group)，记作 $U(n)$。其中行列式为 $1$ 的酉矩阵组成$n$ 阶特殊酉群(special unitary group)，记作 $SU(n)$。", "quote_sha256": "2e5f554f6bc97d5e35152f4c98f831673223f0b35100793852e0ff7ea861fbb1", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "en/chapter-02/03-matrix-groups", "lang": "en", "source_category": "original_transcription", "basis": "registered-book-edition-section-language-context; 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human-review-not-performed", "translation_review": "not-performed; inherited-source-status" }, "result": { "overview_claims": [ { "id": "matrix-adjoint-definition", "text": "复矩阵的伴随矩阵是先转置、再取复共轭所得的共轭转置：$\\mathsf{A}^{\\dagger}=\\overline{\\mathsf{A}^{T}}$", "evidence_ids": [ "e-13affea24d7c1e96c7c79f17" ] }, { "id": "matrix-adjoint-components", "text": "若原矩阵分量为 $a_{ij}$，伴随矩阵相应分量由原矩阵转置位置上的复共轭给出：$a_{ij}^{\\dagger}=\\overline{a_{ji}}$", "evidence_ids": [ "e-13affea24d7c1e96c7c79f17" ] }, { "id": "matrix-adjoint-unitary", "text": "材料用这一运算定义酉矩阵：$n \\times n$ 复矩阵 $\\mathsf{U}$ 满足以下关系时为酉矩阵：\n\n$$\n\\mathsf{U}\\mathsf{U}^{\\dagger}=\\mathsf{I},\\tag{2.12}\n$$\n\n", "evidence_ids": [ "e-13affea24d7c1e96c7c79f17" ] } ], "author_views": [ { "id": "szekeres-adjoint-matrix", "book_id": "szekeres", "edition_id": "source-snapshot-bc3404156f6e", "summary": "作者在矩阵群一节定义复矩阵的伴随为共轭转置，并以 $\\mathsf{U}\\mathsf{U}^{\\dagger}=\\mathsf{I}$ 描述酉矩阵。", "conditions": "讨论对象是有限阶复矩阵；定义本身不要求矩阵可逆。酉矩阵条件另要求乘积等于单位矩阵。", "notation": "矩阵转置记为 $\\mathsf{A}^{T}$，复共轭记为横线，伴随记为 $\\mathsf{A}^{\\dagger}$。", "evidence_ids": [ "e-13affea24d7c1e96c7c79f17", "e-b31e248b6d3d93d7515877b4" ] } ], "reading_suggestions": [ { "id": "matrix-adjoint-reading-next", "text": "先复习第 2.3 节矩阵转置的乘积公式与复共轭记号，再读例 2.15 中共轭转置定义及其用于刻画酉矩阵的例子；阅读范围保持在有限阶复矩阵运算。", "target_ids": [ "e-13affea24d7c1e96c7c79f17", "e-b31e248b6d3d93d7515877b4" ], "evidence_ids": [ "e-13affea24d7c1e96c7c79f17", "e-b31e248b6d3d93d7515877b4" ] } ], "assessment": { "composition": "complete", "key_material": "available", "critical_gaps": [], "whole_book_coverage": "not-exhausted" }, "gaps": [ "收录材料给出了定义和酉矩阵应用；未讨论更广泛的矩阵运算应用。" ] }, "evidence": { "e-13affea24d7c1e96c7c79f17": { "route": "zh/chapter-02/03-matrix-groups", "anchor": "", "statement_id": null, "label": "Szekeres · §2.3", "quote_sha256": "39641e012a264c3bb6da5de47efd714b47f4895ff1002e507f0b115c61f124c5" }, "e-b31e248b6d3d93d7515877b4": { "route": "en/chapter-02/03-matrix-groups", "anchor": "", "statement_id": null, "label": "Szekeres · §2.3", "quote_sha256": "55a9375fa76b75db53b1d4027738c2a9848723bd7d3a7525e8afc6f5ebeff0fc" } }, "candidate_dependencies": [], "input_sha256": "ce074b1052d44a1c814aa4043aad3128948d4f65dd40115314cca045a3e506f2", "archive_sha256": "4bc3fc4549fd07156491939a9dee5c7d249278b829e5e8710adf7302cbf6fdc5" } ] }