Is Rellich's function valued theorem valid for a rank defficient function valued matrix? Theorem (Rellich). Let $\boldsymbol{A}(t) : \mathbb{R}\rightarrow\mathbb{C}^{n \times n}$ be a Hermitian matrix function that depends
on $t$ analytically.
(i) The $n$ roots of the characteristic polynomial of $\boldsymbol{A}(t)$ can be arranged so that each root $\lambda_j(t)$ for $j = 1,\cdots,n$ is an analytic function of $t$.
(ii) There exists an eigenvector $v_j(t)$ associated with $\lambda_j(t)$ for $j = 1,\cdots,n$ satisfying
(1) $(\lambda_j(t)\boldsymbol{I} - \boldsymbol{A}(t))v_j(t)=0\forall t\in\mathbb{R}$
(2) $||v_j(t)||_2=1~\forall t\in\mathbb{R}$
(3) $v_j^*(t)v_k(t)=0~\forall t\in\mathbb{R}$ for $k\neq j$
(4) $v_j(t)$ is an analytic function valued vector of $t$.
My question is what happens if some of the eigenvalues have multiplicity,i.e. $\lambda_j(t)=\lambda_k(t)$ for some values of $t$ if $k\neq j$. I especially need to know whether (i) and (ii)-2 and (ii)-4 remains valid?
 A: Yes, Rellich's theorem does not require the eigenvalues to be distinct.  See e.g. Reed and Simon, "Methods of modern mathematical physics vol. 4: Analysis of Operators", Chapter XII (in particular Problems XII.16 and XII.17).
A: A comment to Robert's answer. Rellich's theorem is actually a simple consequence of the Schwarz symmetry principle from analytic function theory. Take a semi-neighborhood $D$ of
the real axis of the shape $0<y<h(x)$, where $h$ is a positive continuous function, so small that there are no singularities of the algebraic function $\lambda(t)$ in $D$.
(Algebraic functions have only finitely many singularities, so there is no problem with
existence of such $h$). In $D$ the algebraic function breaks into finitely many holomorphic
branches. Each branch must have a real limit when $t$ approaches the real axis.
Because all eigenvalues of an Hermitean matrix are real. Therefore, by the Schwarz symmetry
principle, all branches of $\lambda$ have analytic continuations to a full
neighborhood of the real line. The statements about eigenvectors follow from analyticity of
$\lambda$.
When some $\lambda_j$ collide on the real line, their graphs simply cross, each remaining
analytic. This is called "level crossing" in physics.
