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)

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:

Wolfram Research (2003), CharacteristicPolynomial, Wolfram Language function, https://reference.wolfram.com/language/ref/CharacteristicPolynomial.html (updated 2007).

Text

Wolfram Research (2003), CharacteristicPolynomial, Wolfram Language function, https://reference.wolfram.com/language/ref/CharacteristicPolynomial.html (updated 2007).

BibTeX

@misc{reference.wolfram_2020_characteristicpolynomial, author="Wolfram Research", title="{CharacteristicPolynomial}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/CharacteristicPolynomial.html}", note=[Accessed: 19-January-2021 ]}

BibLaTeX

@online{reference.wolfram_2020_characteristicpolynomial, organization={Wolfram Research}, title={CharacteristicPolynomial}, year={2007}, url={https://reference.wolfram.com/language/ref/CharacteristicPolynomial.html}, note=[Accessed: 19-January-2021 ]}

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