I'm looking for a proof in the literature of the following fact: Let $A_t$ be a $C^1$-function of one argument $t \in (a,b)$ taking values in the self-adjoint $N \times N$ matrices. Suppose that for every $t$ the spectrum of $A_t$ is simple. Denote by $\lambda_t$ a continuous parametrization of an eigenvalue of $A_t$. Then there exists a $C^1$ function $\psi_t$ taking values in $\mathbb{C}^N$ such that $$ A_t \psi_t = \lambda_t \psi_t $$ and $\psi_t$ is normalized $\sum_{j=1}^{N} |\psi_t(n)|^2 = 1$.

It is not hard to give a proof, but it is somewhat awkward to write down, that's why I am looking for a reference of this fact, I couldn't find one in Reed--Simon 4 and Kato. Here is a short sketch of my idea how to proof it: $\psi_t$ lives in $S^N$, whose tangent space at the point $\psi_t$ is $T_{\psi_t} S^N \cong \{\psi_t\}^{\perp}$. Next, we have that $A_t - \lambda_t|_{\psi_t ^{\perp}}$ is invertible (here the simplicity of the spectrum is used). Using this we can define $\psi_t$ as the solution of the ODE $$ \dot{\psi_t} = (A_t - \lambda_t|_{\psi_t ^{\perp}})^{-1} (\dot{A}_t - \dot{\lambda}_t) \psi_t. $$ Now, various computations show that everything is well-defined and indeed give the desired solution.

`$A_t$`

is simple." ;) In any event, for computational work with unsymmetric matrices with possibly multiple eigenvalues, it is best to work with the Schur decomposition. – J. M. Aug 25 '10 at 15:30