I am looking for an appropriate reference for the following fact. I already posted on [math.stackexchange][2], but got no answer.

> For each $X \in \mathbb{R}^{n \times n}_{\text{sym}}$ (symmetric matrix),
> there exist $\varepsilon, L > 0$, such that
> for all $H \in \mathbb{R}^{n \times n}_{\text{sym}}$ with $\|H\| \le \varepsilon$
> the following holds:
> 
> There are orthogonal matrices $P, Q$, such that
> \begin{equation*}
	Q^\top \, X \, Q = \Lambda(X)
\end{equation*}
> and
> \begin{equation*}
	P^\top \, (X + H) \, P = \Lambda(X + H)
\end{equation*}
> and
> \begin{equation*}
	\| P - Q \| \le L \, \| H \|.
\end{equation*}

Here, $\Lambda(X)$ is the diagonal matrix containing the eigenvalues of $X$.
That is, $P$ and $Q$ are bases of eigenvectors of $X$ and $X + H$, respectively.
Here, it is crucial that we can choose $Q$ in dependence of $H$.


This result can be found in 
[this article][1], see Lemma 4.3.
I feel, however, that a reference from 2003 is not appropriate for this "simple" fact.

  [1]: http://dx.doi.org/10.1137/S0036142901393814
  [2]: https://math.stackexchange.com/questions/1133071/reference-continuity-of-eigenvectors