## 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.

\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}

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