Yah, in short, any matrix operation where you can track the effect off the operation on the eigenvalues of the matrix will give you the same effect of polynomia

Yah, in short, any matrix operation where you can track the effect off the operation on the eigenvalues of the matrix will give you the same effect of polynomial roots. I used to work in algebraic combinatorics, which provides maps between important objects in algebra, combinatorics, and geometry, allowing us to apply theorems across the domains. You would often have dead obvious observations on one side which were very difficult to prove on another; eg 'This particular eigenvalue is a positive integer because it is also the dimension of a vector space."