Define $W$ a vector spaces. Let $\alpha(t):W\to W$ be a differentiable linear mapping. Furthermore, let $\lambda_1 (t), \lambda_2 (t), \cdots$ denote the eigenvalues of $\alpha(t)$.
Intuitively, I am guessing $\lambda'_1 (t), \lambda'_2 (t), \cdots$ will be the eigenvalues of $\alpha'(t)$. If this statement is true, could you give me a hint on how to prove it? If it is not true, please point it out.