每週問題 September 21, 2015

Q 是正交矩陣,則 \displaystyle\frac{dQ}{dt}Q^T 是反對稱矩陣。

Let Q=[q_{ij}(t)] be an orthogonal matrix, where each entry q_{ij}(t) is a differentiable function of t. Show that \displaystyle\frac{dQ}{dt}Q^T is skew-symmetric.

 
參考解答:

因為 QQ^T=I,可得

\displaystyle  \begin{aligned}  0&=\frac{dI}{dt}=\frac{d(QQ^T)}{dt}=\frac{dQ}{dt}Q^T+Q\frac{dQ^T}{dt}\\  &=\frac{dQ}{dt}Q^T+Q\left(\frac{dQ}{dt}\right)^T=\frac{dQ}{dt}Q^T+\left(\frac{dQ}{dt}Q^T\right)^T.  \end{aligned}

所以,\displaystyle\left(\frac{dQ}{dt}Q^T\right)^T=-\frac{dQ}{dt}Q^T,即證明所求。

相關閱讀:
Advertisements
This entry was posted in pow 內積空間, 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