每週問題 April 10, 2017

計算 I-\mathbf{x}\mathbf{y}^T 的伴隨矩陣。

Let \mathbf{x} and \mathbf{y} be n-dimensional column vectors. Prove that

\hbox{adj}(I-\mathbf{x}\mathbf{y}^T)=\mathbf{x}\mathbf{y}^T+(1-\mathbf{y}^T\mathbf{x})I.

參考解答:

因為 \mathbf{x} (\mathbf{y}^T\mathbf{x})\mathbf{y}^T=\mathbf{x}\mathbf{y}^T(\mathbf{y}^T\mathbf{x})

\displaystyle \begin{aligned} (I-\mathbf{x}\mathbf{y}^T)\left(\mathbf{x}\mathbf{y}^T+(1-\mathbf{y}^T\mathbf{x})I\right) &=\mathbf{x}\mathbf{y}^T-\mathbf{x}\mathbf{y}^T\mathbf{x}\mathbf{y}^T+(1-\mathbf{y}^T\mathbf{x})I-\mathbf{x}\mathbf{y}^T+\mathbf{x}\mathbf{y}^T\mathbf{y}^T\mathbf{x}\\ &=(1-\mathbf{y}^T\mathbf{x})I.\end{aligned}

使用 Sylvester 行列式定理

\displaystyle \det(I-\mathbf{x}\mathbf{y}^T)=1-\mathbf{y}^T\mathbf{x}

根據伴隨矩陣恆等式 X(\hbox{adj}X)=(\det X)I 即得證。

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2 Responses to 每週問題 April 10, 2017

  1. Wenchao Deng says:

    老师,题目的最后有笔误:不是1-yxT,应该是1-yTx

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