# CharacteristicPolynomial

gives the characteristic polynomial for the matrix m.

CharacteristicPolynomial[{m,a},x]

gives the generalized characteristic polynomial with respect to a.

# Details • m must be a square matrix.
• It can contain numeric or symbolic entries.
• is essentially equivalent to Det[m-id x] where id is the identity matrix of appropriate size. »
• CharacteristicPolynomial[{m,a},x] is essentially Det[m-a x]. »

# Examples

open allclose all

## Basic Examples(1)

Find the characteristic polynomial of a symbolic matrix:

Compare with a direct computation:

## Scope(13)

### Basic Uses(5)

Find the characteristic polynomial of a machine-precision matrix:

Arbitrary-precision polynomial:

Characteristic polynomial of a complex matrix:

An exact characteristic polynomial:

Visualize the result:

The characteristic polynomials of large numerical matrices are computed efficiently:

### Generalized Eigenvalues(4)

The generalized characteristic polynomial :

A generalized machine-precision characteristic polynomial:

Find a generalized exact characteristic polynomial:

The absence of an term indicates an infinite generalized eigenvalue:

Compute the result at finite precision:

Find the generalized characteristic polynomial of symbolic matrices:

### Special Matrices(4)

Characteristic polynomial of sparse matrices:

Characteristic polynomials of structured matrices:

The characteristic polynomial IdentityMatrix is a binomial expansion:

Characteristic polynomial of HilbertMatrix:

## Applications(1)

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

## Properties & Relations(5)

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 (CayleyHamilton 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: