# Cayley-Hamilton theorem I

This post proves Cayley-Hamilton using invariant subspaces.

The Cayley-Hamilton theorem is one of the most basic results in linear algebra.

(Cayley-Hamilton)If is a matrix with entries in a field , is its characteristic polynomial, then .

**Proof**

To show that , it suffices to show that for any vector , . This prompts us to think about invariant subspaces.

Take any . Let be a subspace of . If , it means that the minimal polynomial of is of degree . Let this polynomial be .

Extend a basis of , namely to a basis of . Then the upper block of the matrix wrt the new basis is exactly

Since the characteristic polynomial is an invariant under base change, and by the product formula of determinant of block matrices, we have . It remains to show that , which is easily checked by direct computation.

Since , we have . Since is arbitrary, we have shown that .

**Remark**

- The above proof shows that Cayley-Hamilton holds in any field . According to the wikipedia entry, this theorem is in fact the source of Nakayama’s lemma.