For two matrices $A$ and $B$ over a field $k$, they are similar if and only if their corresponding characteristic matrices $λI−A$ and $λI−B$ are equivalent. Is there an intuitive explanation behind this phenomenon? I have some considerations:
The characteristic matrix $λI−A$ seems to apply a small perturbation to the original matrix $A$. We know that for all $λ$, the matrix $λI−A$ is invertible if and only if $\det (λI-A)$ is nonzero constant. Is there a geometric interpretation for this?
For matrices over the polynomial ring (i.e., polynomial matrices in the form $λI−A$), if we consider them as a form of perturbation, what effects do elementary transformations on matrices over this polynomial ring impose on matrices over $k$?
I have already read the answers in https://math.stackexchange.com/a/2824325/846766, but I am looking for a deeper understanding of the underlying reasons.