We have the following theorems!
Corollary VI.1.6 [Bhatia: Matrix Analysis]: Let $a_j(t)$, where $1\leq j \leq n$ be continuous complex valued functions defined on an interval $I$. Then there exists continuous functions $\alpha_1(t), \ldots, \alpha_n(t)$ such that for each $t \in I$, these constitute the roots of the monic polynomial $z^n-a_1(t)z^{n-1}+\cdots+(-1)^na_n(t)$.
Theorem [Kato: Perturbation theory for linear operators]: Suppose $D \subset \mathbb{C}$, and that $A: D \rightarrow M_n$ is a continuous function. If (1)- D is a real interval, or (2)- A(t) has only real eigenvalues then there exist n eigenvalues (counted with algebraic multiplicities) of A(t) that can be parameterized as continuous functions $\lambda_1(t), \ldots, \lambda_n(t)$ from D to $\mathbb{C}$. In the second case, one can set $\lambda_1(t) \geq \cdots \geq \lambda_n(t)$.
These two theorems imply that the eigenvalues of a matrix (whose entries are continuous functions of a real parameter) are continuous functions of the parameter. Also, the second theorem says that this continuity of eigenvalues is still there if the parameter is a complex number, and the matrix is Hermitian.
Let $A = \left[x_i+x_j\right]$, where $0<x_1<\ldots<x_n$ are positive real numbers. Then the entries of $A$ are the continuous functions of $x_i$'s. Since A is Hermitian, all its eigenvalues are real. The characteristic polynomial of A is a poly with coefficients as functions of $x_i$'s.
Que: Is there any extensions of the above theorems, which can imply that the eigenvalues of $A$ can be parametrized as continuous functions of $x_i$'s?
What I mean is that: Is there any analogous result to Corollary VI.1.6, where $a_j$ and $\alpha_j$ are functions of $x_i$'s?
Is there any analogous result to Kato theorem, where $D \subset \mathbb{R}^n$?
I couldn't find any such result. Any help is really appreciated.
