I am looking for an appropriate reference for the following fact.

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.


1 Answer 1


The "standard" reference for all results of such types is

Kato, T., Perturbation theory for linear operators. Springer-Verlag, 1976

The finite dimensional theory is well exposed in Chapter 2, with the non-analytic dependence in section 5.

As a side remark: your statement is actually not, as your title suggests, about "continuity of eigenvectors", which is false (see Chapter 2, section 5.3 of the above citation). What you are describing is the continuity of eigenprojections.

If you want to be historical, these types of problems were originally treated by Rellich in the 1930s. A more detailed account was provided in his 1954 lecture notes which are freely available.


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