If $M_{n\times n}(\mathbb{C})$ is the space of $n\times n$ matrices with coefficients in the complex numbers I can make a map:
$\phi: M_{n\times n}(\mathbb{C}) \rightarrow Sym^n(\mathbb{C})$
which sends a matrix to its set of eigenvalues. I can show all fibers have the same dimension $(=n^2-n)$ by computing stabilizers of matrices in Jordan form.
Question: Is there a quicker proof or at least a reference for the fact that the fibers of this map have the same dimension?