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20211018 一些特殊矩阵
2022-06-13 08:55:00 【我起个什么名字呢】
酉矩阵(Unitary Matrix): A H A = A A H = I A^HA=AA^H=I AHA=AAH=I,则称酉矩阵(幺正矩阵、么正矩阵)。
正交矩阵:如果酉矩阵的元素都是实数,叫做正交矩阵(正交矩阵都是正数)。 A T A = A A T = I A^TA=AA^T=I ATA=AAT=I。
实对称矩阵:所有元素实数, A T = A A^T=A AT=A。
实反对称矩阵:所有元素实数, A T = − A A^T=-A AT=−A。
厄米特矩阵(Hermitian Matrix):对角线元素实数,非对角线可实可虚, A H = A A^H=A AH=A。特征值一定是实数。
正规矩阵(Normal Matrix): A T A = A A T A^TA=AA^T ATA=AAT,则称为正规矩阵。
任意正规矩阵都可在经过一个酉变换后变为对角矩阵,反过来所有可在经过一个酉变换后变为对角矩阵的矩阵都是正规矩阵。
酉变换:
Schur定理:定理 1.41 (1)设 A ∈ C n × n \boldsymbol{A} \in \mathbf{C}^{n \times n} A∈Cn×n 的特征值为 λ 1 , ⋅ λ 2 , ⋯ , λ n \lambda_{1}, \cdot \lambda_{2}, \cdots, \lambda_{n} λ1,⋅λ2,⋯,λn, 则存 在酉矩阵 P \boldsymbol{P} P,使得
P − 1 A P = P H A P = [ λ 1 ∗ ⋯ ∗ λ 2 ⋱ ⋮ ⋱ ∗ λ n ] \boldsymbol{P}^{-1} \boldsymbol{A P}=\boldsymbol{P}^{\mathrm{H}} \boldsymbol{A} \boldsymbol{P}=\left[\begin{array}{lllc} \lambda_{1} & * & \cdots & * \\ & \lambda_{2} & \ddots & \vdots \\ & & \ddots & * \\ & & & \lambda_{n} \end{array}\right] P−1AP=PHAP=⎣⎢⎢⎢⎡λ1∗λ2⋯⋱⋱∗⋮∗λn⎦⎥⎥⎥⎤
(2)设 A ∈ R n × n \boldsymbol{A} \in \mathbf{R}^{n \times n} A∈Rn×n 的特征值为 λ 1 , λ 2 , ⋯ , λ n \lambda_{1}, \lambda_{2}, \cdots, \lambda_{n} λ1,λ2,⋯,λn, 且 λ i ∈ R ( i = 1 \lambda_{i} \in \mathbf{R}(i=1 λi∈R(i=1, 2 , ⋯ , n ) 2, \cdots, n) 2,⋯,n), 则存在正交矩阵 Q Q Q, 使得
Q − 1 A Q = Q T A Q = [ λ 1 ∗ ⋯ ∗ λ 2 ⋱ ⋮ ⋱ ∗ λ n ] \boldsymbol{Q}^{-1} \boldsymbol{A} \boldsymbol{Q}=\boldsymbol{Q}^{\mathrm{T}} \boldsymbol{A} \boldsymbol{Q}=\left[\begin{array}{lllc} \lambda_{1} & * & \cdots & * \\ & \lambda_{2} & \ddots & \vdots \\ & & \ddots & * \\ & & & \lambda_{n} \end{array}\right] Q−1AQ=QTAQ=⎣⎢⎢⎢⎡λ1∗λ2⋯⋱⋱∗⋮∗λn⎦⎥⎥⎥⎤
定理 1.42 (1)设 A ∈ C n × n \boldsymbol{A} \in \mathbf{C}^{n \times n} A∈Cn×n, 则 A \boldsymbol{A} A 酉相似于对角矩阵的充要 条件是 A \boldsymbol{A} A 为正规矩阵;
(2)设 A ∈ R n × n \boldsymbol{A} \in \mathbf{R}^{n \times n} A∈Rn×n, 且 A \boldsymbol{A} A 的特征值都是实数,则 A \boldsymbol{A} A 正交相似于对 角矩阵的充要条件是 A \boldsymbol{A} A 为正规矩阵.
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