Does Cayley Hamilton Theorem apply for non-diagonalizable matrices as well?

Question

Cayley Hamilton Theorem says that a matrix $A$ satisfies its characteristic equation. My professor proved this for diagonalizable matrices. What happens if $A$ is not diagonalizable? Does the C-H Theorem still hold? Can you give a proof why it holds or why it does not?

Thanks!

Answer 1

It holds in general. Also the proof can be found in https://en.wikipedia.org/wiki/Cayley%E2%80%93Hamilton_theorem