每週問題 June 13, 2016

本週問題是推導兩個座標系統的變換矩陣。

Let \boldsymbol{\beta}=\{\mathbf{x}_1,\ldots,\mathbf{x}_k\} and \boldsymbol{\gamma}=\{\mathbf{y}_1,\ldots,\mathbf{y}_k\} be bases for a subspace \mathcal{V} in \mathbb{R}^n. Let X=\begin{bmatrix} \mathbf{x}_1&\cdots&\mathbf{x}_k \end{bmatrix} and Y=\begin{bmatrix} \mathbf{y}_1&\cdots&\mathbf{y}_k \end{bmatrix}. Show that the change of coordinates matrix from \boldsymbol{\beta} to \boldsymbol{\gamma} is

P=(Y^TY)^{-1}Y^TX.

 
參考解答:

對於任意 \mathbf{v}\in\mathcal{V},設參考基底 \boldsymbol{\beta}\boldsymbol{\gamma} 的座標向量分別為

[\mathbf{v}]_{\boldsymbol{\beta}}=\begin{bmatrix} c_1\\ \vdots\\ c_k\end{bmatrix},~~[\mathbf{v}]_{\boldsymbol{\gamma}}=\begin{bmatrix} d_1\\ \vdots\\ d_k\end{bmatrix}

因此,\mathbf{v}=\sum_{i=1}^kc_i\mathbf{x}_i=X[\mathbf{v}]_{\boldsymbol{\beta}}\mathbf{v}=\sum_{i=1}^kd_i\mathbf{y}_i=Y[\mathbf{v}]_{\boldsymbol{\gamma}}。合併兩式,X[\mathbf{v}]_{\boldsymbol{\beta}}=Y[\mathbf{v}]_{\boldsymbol{\gamma}}。左乘 Y^TY^TX[\mathbf{v}]_{\boldsymbol{\beta}}=Y^TY[\mathbf{v}]_{\boldsymbol{\gamma}}。因為 \hbox{rank}(Y^TY)=\hbox{rank}Y=kk\times k 階矩陣 Y^TY 是可逆的。因此,

[\mathbf{v}]_{\boldsymbol{\gamma}}=(Y^TY)^{-1}Y^TX[\mathbf{v}]_{\boldsymbol{\beta}}

證畢。若 k=n,則 P=(Y^TY)^{-1}Y^TX=Y^{-1}(Y^T)^{-1}Y^TX=Y^{-1}X

Advertisements
This entry was posted in pow 線性變換, 每週問題 and tagged , . Bookmark the permalink.

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