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?



Is Rellich's function valued theorem valid for a rank defficient function valued matrix?
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).


