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.
- CharacteristicPolynomial[m,x] 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 allBasic Examples (1)
Scope (13)
Basic Uses (5)
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:
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 (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
:
Text
Wolfram Research (2003), CharacteristicPolynomial, Wolfram Language function, https://reference.wolfram.com/language/ref/CharacteristicPolynomial.html (updated 2007).
BibTeX
BibLaTeX
CMS
Wolfram Language. 2003. "CharacteristicPolynomial." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2007. https://reference.wolfram.com/language/ref/CharacteristicPolynomial.html.
APA
Wolfram Language. (2003). CharacteristicPolynomial. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/CharacteristicPolynomial.html