Consider an $n \times n$ matrix $A = A(t)$. Denote its eigenvalues by $\lambda_{i}(t)$ and eigenvectors by $v_{i}(t)$.
I have 2 questions:
Q1) What is the rate of change of the eigenvalues?
Q2) What is the rate of change of the eigenvectors (expressed in terms of $v_{i}'s$)?
I am thinking of doing something like starting with:
$A(t)v_{i}(t) = \lambda_{i}(t)v_{i}(t)$
Taking derivative,
$\dot{A}v_{i} + A \dot{v_{i}} = \dot{\lambda} v_{i} + \lambda \dot{v_{i}}$
How can I proceed?