I am looking for an appropriate reference for the following fact. I already posted on math.stackexchange, 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, see Lemma 4.3. I feel, however, that a reference from 2003 is not appropriate for this "simple" fact.

I am looking for an appropriate reference for the following fact. I already posted on math.stackexchange, 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, see Lemma 4.3. I feel, however, that a reference from 2003 is not appropriate for this "simple" fact.