The statement of the Cayley-Hamilton Theorem is fairly straight-forward.
I now know how to find characteristic polynomials from a given matrix (or at least a matrix with certain properties that I am unaware of!).
I know that the eigenvalues of the matrix are roots of the polynomial.
But what does having such a polynomial mean? Wikipedia says that the characteristic polynomial "...encodes several important properties of the matrix...", but once we have switched to "matrix form" of the equation, what can we conclude?
In other words, what does the Theorem do for us, besides allowing us to say, "Hey, I know a matrix solution to this polynomial"?? Is there an abstraction of this in abstract algebra (rings, fields, etc.)?
Thanks for your time.