On the eigenvectors of a matrix-valued function A(t) and its limit I have a question on the following theorem taken from Lax Linear Algebra and its Applications (Thm. 7, Ch. 9.1):

Let A(t) be a differentiable matrix-valued function of t, a(t) an eigenvalue of A(t) of multiplicity one. Then we can choose an eigenvector h(t) of A(t) pertaining to the eigenvalue a(t) to depend differentiably on t.

In the theorem, there are no assumptions on $A(t)$. In my case, $A(t)$ is symmetric and positive-definite for all values of $t$, with eigenvalues in a bounded interval that depends on $t$. I am currently still assuming that $A(t)$ has only simple eigenvalues for all $t$, which is not necessarily the case though.
I am particularly interested in the case $A_\infty = \lim_{t\rightarrow\infty} A(t)$.

*

*Based on the above theorem, is it fair to assume that the eigenvectors of $A(t)$ can be chosen such that they converge to those of $A_\infty$?

In some cases, $A_\infty=0$ and the above is still valid but contains little information on how the eigenvectors of $A(t)$ behave. However,

*

*If $\tilde{A}_\infty = t^3 (A(t)-\frac{c t -1}{t^2}I)$ with some constant $c$, which I can show using the first few terms of its convergent Neumann series, am I able to apply the above theorem to $A(t)$ using the limit $\tilde{A}_\infty$?

Lax also has some results on multiple eigenvalues of self-adjoint matrices, which only hold for $t$ in a neighbourhood of $A(0)$ (or any other point).

*

*Are there any stronger results for symmetric positive-definite matrices with multiple eigenvalues out there? They could also be formulated in terms of orthogonal projections onto eigenspaces...

Thank you very much.
 A: 
Based on the above theorem, is it fair to assume that the eigenvectors of $A(t)$ can be chosen such that they converge to those of $A_\infty$?

Yes. Technically, a bit of work is required if we want to deduce this as a consequence of the theorem. One approach is to define the continuously differentiable auxiliary matrix function $B:[0,\infty) \to \Bbb F^{n \times n}$ given by
$$
B(t) = \begin{cases}
A_\infty & t=0,\\
A(1/t) & t > 0
\end{cases}
$$
and consider the eigenvalue/eigenvectors of $B(t)$ near $t=0$.

If $\tilde{A}_\infty = t^3 (A(t)-\frac{c t -1}{t^2}I)$ with some constant $c$, which I can show using the first few terms of its convergent Neumann series, am I able to apply the above theorem to $A(t)$ using the limit $\tilde{A}_\infty$

Yes.

Are there any stronger results for symmetric positive-definite matrices with multiple eigenvalues out there? They could also be formulated in terms of orthogonal projections onto eigenspaces...

I'm not sure if this is the kind of thing you're interested in, but here's something that you get (I believe it's stated in Kato's "Perturbation theory for linear operators").

Let $A(t)$ be a (real-entry) matrix-valued function with continuous eigenvalue/eigenvector parameterization $\lambda(t),v(t)$ where for each $t$, $v(t)$ is a unit-vector. If $A,v,\lambda$ are continuously differentiable, then
$$
\lambda'(a) = v(a)^\top A'(a) v(a).
$$

