I have a real-valued matrix, $M(a)$, which is a differentiable function of $a$, but not continuously differentiable, with $M(0)=I$. I'll assume $M'(0)$ has distinct eigenvalues.
I'm looking for results proving the differentiability of the eigenvalues of $M(0)$. I suspect it may be a challenging proof because when I read the literature, I see people making very restrictive assumptions, usually that the eigenvalues are distinct (clearly not true for me), and if not, it seems that people assume the matrix is symmetric, also not true for me.
If I assume the eigenvalues and eigenvectors are differentiable, say with $v_1(a)$ being one of the eigenvectors, then I believe I can conclude that $v_1(0)$ is an eigenvector of $M'(0)$ and I can work out $\lambda_1'(0)$. If so I've got the result I'm trying to prove.
Is it known whether the eigenvalues of a differentiable real matrix are in fact differentiable, even if repeated? If true - where can I find a reference? If false, where can I found a counterexample? If unknown - what's the challenge in proving it?