CharacteristicPolynomial

CharacteristicPolynomial[m,x]

gives the characteristic polynomial for the matrix m.

CharacteristicPolynomial[{m,a},x]

gives the generalized characteristic polynomial with respect to a.

Details

Examples

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Basic Examples  (1)

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)