每週問題 November 9, 2015

這是廣義特徵值 (generalized eigenvalue) 問題。

Let A and B be n\times n matrices. If B is nonsingular, show that there is a complex scalar \lambda such that A -\lambda B is singular.

 
參考解答:

若存在一非零向量 \mathbf{x} 使得 (A-\lambda B)\mathbf{x}=\mathbf{0},則 A-\lambda B 為不可逆矩陣,其中 \lambda 稱為 AB 的廣義特徵值。因為 B 可逆,上式左乘 B^{-1},可得 (B^{-1}A-\lambda I)\mathbf{x}=\mathbf{0}。因此,B^{-1}A 的特徵值 \lambda 即為所求。

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This entry was posted in pow 特徵分析, 每週問題 and tagged . Bookmark the permalink.

1 Response to 每週問題 November 9, 2015

  1. Rongwei Sun says:

    看到定義說實對稱矩陣,複矩陣求特徵值的問題,一直困擾我,懇請您指教。

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