矩阵乘积的特征值等于特征值的乘积(Conditioned)

当矩阵 \(A\) 和 \(B\) 有相同的特征向量时, \(AB\)(或\(BA\)) 的特征值等于 \(A\), \(B\) 特征值之积.
Proof
设 \(x\) 是 \(A\) 的关于特征值 \(\lambda\) 的特征向量, 则 \(Ax = \lambda x\), 且 \(x\) 是 \(B\) 的关于特征值 \(\mu\) 的特征向量, 即 \(Bx = \mu x\),
则

\[BAx = B\lambda x = \lambda Bx = (\lambda\mu) x, \]

\[ABx = A\mu x = \mu Ax = (\mu\lambda) x. \]

End Proof.

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当矩阵 \(A\) 和 \(B\) 有相同的特征向量时, \(A\), \(B\) 乘积可交换, 即 \(AB = BA\).
Proof
设 \(AB\) 的 \(n\) 个线性独立的特征向量构成的矩阵为 \(P = [x_1,...,x_n], x_i\in R^n\), 特征值构成的对角矩阵为 \(D\),
则有

\[(AB)P = PD, \rightarrow AB = PDP^{-1}, \]

根据 \(AB\) 与 \(BA\) 有相同的特征值和特征向量(见上述证明), 则

\[(BA)P = PD, \rightarrow BA = PDP^{-1}, \]

则 \(AB=BA\).
End Proof

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例子
设 \(Ax = \lambda x\), 则 \((A-\alpha I)x = (\lambda-\alpha)x\), \((A+\alpha I)x = (\lambda +\alpha)x\),
则

\[(A(A-\alpha I))x = A(\lambda-\alpha)x = (\lambda-\alpha)\lambda x = ((A-\alpha I)A)x \]

\[(A(A+\alpha I))x = A(\lambda+\alpha)x = (\lambda+\alpha)\lambda x = ((A+\alpha I)A)x \]

\[((A-\alpha I)(A+\alpha I))x = (A-\alpha I)(\lambda+\alpha)x = (\lambda+\alpha)(\lambda - \alpha)x = ((A+\alpha I)(A-\alpha I))x \]

此外,
当 \(A-\alpha I\) 可逆时, 有

\[x = (A-\alpha I)^{-1}(A-\alpha I)x = (A-\alpha I)^{-1}(\lambda-\alpha)x, \]

则 \((A-\alpha I)^{-1}x = \frac{1}{(\lambda-\alpha)}x\),
当 \(A+\alpha I\) 可逆时, 有

\[x = (A+\alpha I)^{-1}(A+\alpha I)x = (A+\alpha I)^{-1}(\lambda+\alpha)x, \]

则 \((A+\alpha I)^{-1}x = \frac{1}{(\lambda +\alpha)}x\),

则

\[(A(A-\alpha I)^{-1})x = A\frac{1}{\lambda -\alpha}x = \frac{\lambda}{\lambda -\alpha}x \]

\[(A(A+\alpha I)^{-1})x = A\frac{1}{\lambda +\alpha}x = \frac{\lambda}{\lambda +\alpha}x \]

\[((A+\alpha I)(A-\alpha I)^{-1})x = (A+\alpha I)\frac{1}{\lambda -\alpha}x = \frac{\lambda+\alpha}{\lambda -\alpha}x \]

\[((A-\alpha I)(A+\alpha I)^{-1})x = (A-\alpha I)\frac{1}{\lambda +\alpha}x = \frac{\lambda-\alpha}{\lambda +\alpha}x \]