# Characteristic polynomial

From Citizendium

In linear algebra the **characteristic polynomial** of a square matrix is a polynomial which has the eigenvalues of the matrix as roots.

Let *A* be an *n*×*n* matrix. The characteristic polynomial of *A* is the determinant

where *X* is an indeterminate and *I*_{n} is an identity matrix.

The characteristic polynomial is unchanged under similarity, and hence be defined for an endomorphism of a vector space, independent of choice of basis.

## Properties

- The characteristic polynomial is monic of degree
*n*; - The set of roots of the characteristic polynomial is equal to the set of eigenvalues of
*A*.

## Cayley-Hamilton theorem

The **Cayley-Hamilton theorem** states that *a matrix satisfies its own characteristic polynomial*.