{"schema_version":"ocme-site-results-v1","locale":"zh-Hant","page_type":"derivations","facts":{"questions":36000,"reviewed":36000,"display_candidates":0,"knowledge":302,"guides":299,"paths":59,"batches":360,"families":1441,"atlas":302,"core":80,"evidence":9,"formal_evidence":5,"formal_subjects":4,"lean_files":6,"lean_toolchain":"leanprover/lean4:v4.30.0","question_state":"candidate","english_state":"ai_candidate","question_Lean":"NOT_RUN","canonical_truth_promotion":false},"filters":{"q":"mko-eigenvalue"},"kind":"guides","total":1,"pages":1,"page":1,"entries":[{"id":"derivation-mko-eigenvalue-01","title":{"zh-Hant":"二階實矩陣的特徵值：特徵多項式、判別式與非零特徵向量","en":"Eigenvalues of 2×2 real matrices: characteristic polynomial, discriminant and nonzero eigenvectors"},"summary":{"zh-Hant":"對 2×2 實矩陣證明：λ 是實特徵值 ⇔ det(M−λI)=0 ⇔ λ²−tr(M)λ+det(M)=0，且互異實特徵值的個數由 Δ=tr(M)²−4det(M) 的正負決定（負、零、正分別為 0、1、2）。以 M=[[4,1],[2,3]] 求出全部互異實特徵值與各自的非零特徵向量，說明候選 λ=3 的核 K₃={0} 而 3 不是特徵值，並拒絕候選特徵對 (3,0)；再以旋轉矩陣 R=[[0,−1],[1,0]]（Δ<0）與 J=[[3,1],[0,3]]（Δ=0）對照。","en":"For 2×2 real matrices prove: λ is a real eigenvalue ⇔ det(M−λI)=0 ⇔ λ²−tr(M)λ+det(M)=0, and the number of distinct real eigenvalues is decided by the sign of Δ=tr(M)²−4det(M) (negative, zero, positive give 0, 1, 2). For M=[[4,1],[2,3]] find all distinct real eigenvalues with nonzero eigenvectors, show that for the candidate λ=3 the kernel is K₃={0} and 3 is not an eigenvalue, and reject the candidate eigenpair (3,0); then contrast with the rotation R=[[0,−1],[1,0]] (Δ<0) and J=[[3,1],[0,3]] (Δ=0)."},"mko_ids":["mko-eigenvalue"]}],"title":"概念推導導讀"}