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?