I'm working on the following problem: Let $A(z)$ be an $n \times n$ matrix whose entries $A_{ij}(z)$ are polynomials in $z$. Let $\lambda_j(z), j = 1,\dots,n$ be the eigenvalues of $A(z)$. If $\lambda_j(z_0)$ is a simple eigenvalue, show that $\lambda_j(z)$ is holomorphic in a neighborhood of $z_0$. If $\lambda_j(z_0)$ has multiplicity $p$, show that there is a representation $\lambda_j(z) = \sum_{k=0}^{\infty} c_k (z - z_0)^{\frac{k}{l}}$ for some $1 \leqslant l \leqslant p$.
Let $P(z,w) = \det (A(z) - wI)$, then $P$ is a polynomial in $z,w$. If $\lambda_j(z_0)$ is simple, then $P(z,\lambda_j(z_0)) = 0$ and $\frac{\partial P}{\partial w}(z_0, \lambda_j(z_0)) \neq 0$. By the implicit function theorem, $\lambda_j(z)$ is holomorphic in a neighborhood of $z_0$. For the second part, I'm trying to apply the IFT to the $(p-1)$-th derivative of $P$, but can't argue why $\lambda_j(z)$ admits the desired power series representation.
Any insight would be appreciated.
