每週問題 May 1, 2017

證明嚴格對角佔優 (strictly diagonally dominant) 矩陣是可逆矩陣。

Let A=[a_{ij}] be an n\times n matrix. Prove that if |a_{ii}|>\sum_{j\neq i}|a_{ij}| for i=1,\ldots,n, then A is invertible.

 
參考解答:

使用逆否命題法。假設存在 \mathbf{x}\neq\mathbf{0} 使得 A\mathbf{x}=\mathbf{0},即 A 是不可逆的。設 x_k 為向量 \mathbf{x} 的最大元,\vert x_k\vert\ge\vert x_j\vertj\neq k。乘開 A\mathbf{x}=\mathbf{0} 並取其第 k 元,

a_{k1}x_1+a_{k2}x_2+\cdots+a_{kn}x_n=0

將第 k 項抽離出來,

\displaystyle a_{kk}x_k=-\sum_{j\neq k}a_{kj}x_j

等號左右取絕對值,使用三角不等式可得

\begin{aligned}\displaystyle  \vert a_{kk}\vert\cdot\vert x_k\vert&=\left|\sum_{j\neq k}a_{kj}x_j\right|\le\sum_{j\neq k}\vert a_{kj}x_j\vert\\ &=\sum_{j\neq k}\vert a_{kj}\vert\cdot\vert x_j\vert\\ &\le\left(\sum_{j\neq k}\vert a_{kj}\vert\right)\vert x_k\vert.\end{aligned}

上面最後一個步驟使用已知條件 \vert x_k\vert\ge\vert x_j\vertj\neq k。消除 \vert x_k\vert 後即得

\displaystyle  \vert a_{kk}\vert\le\sum_{j\neq k}\vert a_{kj}\vert

我們得到與嚴格對角佔優矩陣互斥的結果,故得證。

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