Use exact arithmetic to find the characteristic polynomial:

Use machine arithmetic:

Use 20‐digit precision arithmetic:

The characteristic polynomial of a complex matrix:

The generalized characteristic polynomial :

Find the eigenvalues of a matrix as the roots of the characteristic polynomial:

The characteristic polynomial is equivalent to Det[m - id x]:

The generalized characteristic polynomial is equivalent to Det[m - a x]:

A matrix is a root of its characteristic polynomial (Cayley–Hamilton theorem [more...]):

Evaluate the polynomial at m with matrix arithmetic:

Use the more efficient Horner's method to evaluate the polynomial:

where are the eigenvalues is equivalent to the characteristic polynomial:

If is a monic polynomial, then the characteristic polynomial of its companion matrix is:

Form the companion matrix: