# 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

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## Scope(2)

Use exact arithmetic to find the characteristic polynomial:

Use machine arithmetic:

Use 20digit precision arithmetic:

The characteristic polynomial of a complex matrix:

## Generalizations & Extensions(1)

The generalized characteristic polynomial :

## 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:

Introduced in 2003
(5.0)
|
Updated in 2007
(6.0)