{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-eigenvalue",
  "semantic_family_id": "sf-eigenvalue",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "實二階方陣的特徵值與非零特徵向量"
  },
  "summary": {
    "zh-Hant": "特徵向量須非零；候選特徵空間核含零向量，實二階矩陣可沒有實特徵值。"
  },
  "statement": {
    "zh-Hant": "對2×2實矩陣 M，實數 λ 是特徵值當且僅當存在非零 v∈R² 使 Mv=λv。本輪採非零特徵向量慣例；候選 λ 的核 Kλ={v:Mv=λv} 仍包含零向量，即使 λ 並非特徵值。互異實特徵值是 λ²−tr(M)λ+det(M)=0 的互異實根。"
  },
  "formula": {
    "tex": "E(M)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>E</mi><mo>(</mo><mi>M</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "E",
        "glyph": "E"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "M",
          "glyph": "M"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "abe325ab11a820eba562f9fb4b5e700bb14d5f0c971c96a741eb04f12549dc0c"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "全程在實數域與二階方陣，不聲稱每個實矩陣都有實特徵值。"
    },
    {
      "id": "a2",
      "zh-Hant": "候選 eigenpair 題允許零向量輸入，但答案須否；kernel membership 題則允許並接受零向量。"
    },
    {
      "id": "a3",
      "zh-Hant": "判別式 Δ=tr(M)²−4det(M) 為負、零、正時，互異實特徵值數分別為0、1、2；不把重根按代數重數算兩個。"
    }
  ],
  "symbols": [
    {
      "token": "E",
      "role": "real_eigenvalue_summary",
      "role_zh": "本輪實特徵值摘要",
      "scope": "representative_formula"
    },
    {
      "token": "M",
      "role": "real_two_by_two_matrix",
      "role_zh": "2×2實方陣",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-linear-transformation",
      "title_zh": "矩陣表示的實平面線性映射",
      "reason_zh": "Mv 的線性模型。"
    },
    {
      "id": "mko-determinant",
      "title_zh": "實二階方陣的行列式",
      "reason_zh": "特徵多項式的常數係數。"
    },
    {
      "id": "mko-quadratic-equation",
      "title_zh": "二次方程式",
      "reason_zh": "互異實根與判別式切片。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "Mv=λv 對 v=0 自動成立，不能用它讓每個 λ 都成為特徵值。非零向量條件與核會員條件須分開。",
      "對角矩陣 diag(a,d) 的互異實特徵值集合是 {a,d}；a=d 時只列一次。這不聲稱重數、特徵基底或 Jordan 結構已建立。",
      "對一般實二階矩陣只精確算判別式與互異根數；正判別式可以不是完全平方，仍有兩個互異實根，無須用浮點值輸出根。本輪不做一般對角化或複數譜。"
    ],
    "common_misconception_zh": "把零向量當特徵向量、把核中有零向量當成特徵值存在，或把二重根算成兩個互異值。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "實二階整數切片可重算；一般譜、對角化、代數重數及特徵空間理論仍未附形式 Evidence。",
    "evidence_refs": [],
    "producers": [
      {
        "id": "lean-mathlib",
        "adapter": "formal_system",
        "config_path": "formal/lean/lakefile.toml",
        "status": "configured"
      }
    ]
  },
  "formalization": {
    "status": "not_formalized",
    "target_systems": [
      "Lean 4",
      "Mathlib"
    ],
    "next_obligation_zh": "依固定版 Module.End.HasEigenvector／eigenspace 綁定非零條件及2×2特徵多項式橋；保留實根論域。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-eigenvalue.md 整理；2026-10-06 脈衝 R04（使用者手動接續 2026-10-07 工作序列；實際日期仍為 2026-10-06），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-06"
  }
}