每週問題 June 12, 2017

證明 Gram-Schmidt 正交化定理。

Let $\mathbf{v}_1,\ldots,\mathbf{v}_n$ be a basis of an inner product space. Prove that there exists an orthogonal basis $\mathbf{e}_1,\ldots,\mathbf{e}_n$ such that $\mathbf{e}_i\in\hbox{span}\{\mathbf{v}_1,\ldots,\mathbf{v}_i\}$ for all $i=1,\ldots,n$ .

參考解答：

使用數學歸納法證明。若 $n=1$ ，設 $\mathbf{e}_1=\mathbf{v}_1$ ，命題顯然成立。假設存在一組兩兩正交的向量集 $\mathbf{e}_1,\ldots,\mathbf{e}_k$ 使得 $\mathbf{e}_i\in\hbox{span}\{\mathbf{v}_1,\ldots,\mathbf{v}_i\}$ ， $i=1,\ldots,k$ 。考慮

$\mathbf{e}_{k+1}=c_1\mathbf{e}_1+\cdots+c_k\mathbf{e}_k+\mathbf{v}_{k+1}$ 。

我們選擇 $c_i$ 使得 $\left\langle\mathbf{e}_i,\mathbf{e}_{k+1}\right\rangle=0$ ，即 $c_i\left\langle\mathbf{e}_i,\mathbf{e}_i\right\rangle+\left\langle\mathbf{e}_i,\mathbf{v}_{k+1}\right\rangle=0$ ，解得

$\displaystyle c_i=-\frac{\left\langle\mathbf{e}_i,\mathbf{v}_{k+1}\right\rangle} {\left\langle\mathbf{e}_i,\mathbf{e}_i\right\rangle},~~~ i=1,\ldots,k$ 。

因為 $\mathbf{v}_{k+1}\notin\hbox{span}\{\mathbf{v}_1,\ldots,\mathbf{v}_k\}=\hbox{span}\{\mathbf{e}_1,\ldots,\mathbf{e}_k\}$ ，推論 $\mathbf{e}_{k+1}\neq\mathbf{0}$ ，證明 $\mathbf{e}_1,\ldots,\mathbf{e}_{k+1}$ 為兩兩正交的向量集。

1 Response to 每週問題 June 12, 2017

Meiyue Shao says:

06/13/2017 at 9:28 am

其实用 Cholesky 分解来证明也挺方便的

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