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。

Advertisements
This entry was posted in pow 內積空間, 每週問題 and tagged . Bookmark the permalink.

One Response to 每週問題 June 12, 2017

  1. Meiyue Shao says:

    其实用 Cholesky 分解来证明也挺方便的

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google+ photo

You are commenting using your Google+ account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s