This question is quite contrived and general, but I would like to know if there are any interesting visual relationships between a square matrix and its characteristic polynomial. By "any", I mean any image involving a graphical interpretation of a matrix and its characteristic polynomial that makes some sort of visually intuitive sense.
I was surprised to see that searching "characteristic polynomial visualization" online makes it seems as though nobody has thought of this (perhaps this is because it is useless, but it's nice to know anyways).
For example, for any $2\times2$ real square matrix with characteristic polynomial $C_A(\lambda)$, what would the set of points (represented as vectors) $\left\{A[x\space\space C_A(x)]^T,x\in\mathbb{R}\right\}$ look like on the 2D cartesian plane (basically, we transform every point in the plot of $C_A(\lambda)$ by $A$)?
Given $A=\begin{bmatrix}1&2\\3&4\end{bmatrix}$ and $C_A(\lambda)=\lambda^2-5\lambda-2$, we have
Where green represents the original plot of $C_A(\lambda)$ and black is the plot of $C_A(\lambda)$ transformed by $A$.
The $2\times2$ identity matrix intuitively does not change its characteristic polynomial graph.
The matrix $A=\begin{bmatrix}1&1\\1&1\end{bmatrix}$ with $C_A(\lambda)=\lambda^2-2\lambda$ interestingly produces a line which ends abruptly.
These are just examples, though. The more general question still stands.