 # Let A be the set of all n &#x00D7;<!-- × --> n symmetric real matirix and f Willow Pratt 2022-07-12 Answered
Let $A$ be the set of all $n×n$ symmetric real matirix and $f\in C\left(\mathbb{R},A\right)$. Then whether there is a $g\in C\left(\mathbb{R},O\left(n\right)\right)$ such that for all $t\in \mathbb{R}$, $g\left(t{\right)}^{-1}f\left(t\right)g\left(t\right)$ is a diagonal matrix?
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No. Consider $f\left(t\right)=\left(\begin{array}{cc}t& 0\\ 0& 2t\end{array}\right)$ for $t\ge 0$ and $\left(\begin{array}{cc}0& t\\ t& 0\end{array}\right)$.
For $t>0$, $g\left(t\right)$ must be $\left(\begin{array}{cc}±1& 0\\ 0& ±1\end{array}\right)$ or $\left(\begin{array}{cc}0& ±1\\ ±1& 0\end{array}\right)$, but for $t<0$ all entries of $g\left(t\right)$ must be $±1/\sqrt{2}$