Question

Let $A\in\mathcal M_n$ be an $n\times n$ real [symmetric] matrix which depends smoothly on a [finite] set of parameters, $A=A(\xi_1,\ldots,\xi_k)$. We can view it as a smooth function $A:\mathbb R^k\to\mathcal M_n$.

1. What conditions should the matrix $A$ satisfy so that its eigenvalues $\lambda_i(\xi_1,\ldots,\xi_k)$, $i=1,\ldots,n$, depend smoothly on the parameters $\xi_1,\ldots,\xi_k$?

e.g. if the characteristic equation is $\lambda^3-\xi=0$, then the solution $\lambda_1=\sqrt[3] \xi$ is not derivable at $\xi=0$.

2. What additional conditions should the matrix $A$ satisfy so that we can choose a set of eigenvectors $v_i(\xi_1,\ldots,\xi_k)$, $i=1,\ldots,n$, which depend smoothly on the parameters $\xi_1,\ldots,\xi_k$?

Update - important details

• The domain is simply connected
• The rank of $A$ can change in the domain
• The multiplicities of the eigenvalues can change in the domain, they can cross
• The matrix $A$ is real symmetric
• $n$ and $k$ are finite

Update 2

• A relaxation of the conditions of the problem: For fixed $p=(\xi_{01},\ldots,\xi_{0k})$, can we find an open neighborhood of $p$ in the domain and a set of conditions ensuring the smoothness of the eigenvalues and the eigenvectors?