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

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!

• It does indeed hold in general – Aaron Zolotor Nov 2 '18 at 2:22
• Yes, it still holds. One way to see this is to use the fact that diagonalizable matrices are dense in all matrices. – Qiaochu Yuan Nov 2 '18 at 2:22
• @QiaochuYuan, can you please elaborate? – Nagabhushan S N Nov 2 '18 at 2:43