线性代数二级结论

zedo2022年9月24日大约 2 分钟约 499 字

特征值相关

若 $A$ 是三阶矩阵， $A^{*}$ 是伴随矩阵， $tr (A)$ 是 $A$ 的迹，证明：

∣ λ E - A ∣ = λ^{3} - tr (A) λ^{2} + tr (A^{*}) λ - ∣ A ∣

不妨设 $A = a_{11} a_{21} a_{31} a_{12} a_{22} a_{32} a_{13} a_{23} a_{33}$ ，则

D = ∣ A - λ E ∣ = a_{11} - λ a_{21} a_{31} a_{12} a_{22} - λ a_{32} a_{13} a_{23} a_{33} - λ = a_{11} a_{21} a_{31} a_{12} a_{22} - λ a_{32} a_{13} a_{23} a_{33} - λ - λ a_{21} a_{31} 0 a_{22} - λ a_{32} 0 a_{23} a_{33} - λ = a_{11} a_{21} a_{31} a_{12} a_{22} a_{32} a_{13} a_{23} a_{33} - λ - a_{11} 0 a_{31} a_{12} λ a_{32} a_{13} 0 a_{33} - λ - λ a_{22} - λ a_{32} a_{23} a_{33} - λ = ∣ A ∣ - a_{11} a_{21} 0 a_{12} a_{22} 0 a_{13} a_{23} λ - λ a_{11} a_{31} a_{13} a_{33} - λ - λ [a_{22} a_{32} a_{23} a_{33} - λ - λ a_{32} 0 a_{33} - λ] = ∣ A ∣ - λ M_{33} - λ [M_{22} - a_{11} 0 a_{13} λ] - λ [M_{11} - a_{22} 0 a_{23} λ - λ (a_{33} - λ)] = ∣ A ∣ - λ M_{33} - M_{22} λ + a_{11} λ^{2} - λ M_{11} + λ [a_{22} λ + a_{33} λ - λ^{2}] = ∣ A ∣ - (M_{11} + M_{22} + M_{33}) λ + (a_{11} + a_{22} + a_{33}) λ^{2} - λ^{3} = ∣ A ∣ - (A_{11} + A_{22} + A_{33}) λ + tr (A) λ^{2} - λ^{3} = - λ^{3} + tr (A) λ^{2} - tr (A^{*}) λ + ∣ A ∣

故

∣ λ E - A ∣ = (- 1)^{3} ∣ A - λ E ∣ = λ^{3} - tr (A) λ^{2} + tr (A^{*}) λ - ∣ A ∣