# 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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## Basic Examples(3)

Find the characteristic polynomial of a matrix with integer entries:

Visualize the polynomial:

Find the characteristic polynomial in of the symbolic matrix :

Compare with a direct computation:

Compute the characteristic polynomials of the identity matrix and zero matrix:

## Scope(15)

### Basic Uses(7)

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:

Characteristic polynomial of a matrix with finite field elements:

Characteristic polynomial of a CenteredInterval matrix:

Find a random representative mrep of m:

Verify that the coefficients of p contain the coefficients of the characteristic polynomial of mrep:

### 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(6)

Find the characteristic polynomial of the matrix and compare the behavior for , and :

Examining the roots, there is a root at independent of :

For the root at is repeated:

For there are three distinct real roots:

And for , is the only real root, with the other two roots a complex conjugate pair:

Visualize the three polynomials, zooming in on the "bounce" of the plot at the double root :

Compute the determinant of a matrix as the constant term in its characteristic polynomial:

Substitute in :

This result is also the product of the roots of the characteristic polynomial:

Compare with a direct computation using Det:

Compute the trace of a matrix as the coefficient of the subleading power term in the characteristic polynomial:

Extract the coefficient of , where is the height or width of the matrix:

This result is also the sum of the roots of the characteristic polynomial:

Compare with a direct computation using Det:

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

Compare with a direct computation using Eigenvalues:

Use the characteristic polynomial to find the eigenvalues and eigenvectors of the matrices and :

The two matrices have the same characteristic polynomial:

Thus, they will both have the same eigenvalues, which are the roots of the polynomial:

The eigenvectors are given by the null space of :

Eigensystem gives the same result, though it sorts eigenvalues by absolute value:

While has the same eigenvalues as , it has different eigenvectors:

Visualize the two sets of eigenvectors:

Find the generalized eigensystem of with respect to as the roots of the characteristic polynomial:

The roots of the generalized characteristic polynomial are the generalized eigenvalues:

The generalized eigenvectors are given by the null space of :

Compare with a direct computation using Eigensystem:

## Properties & Relations(8)

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:

The sum of the roots of the characteristic polynomial is the trace (Tr) of the matrix:

Similarly, the product of the roots is the determinant (Det):

A matrix and its transpose have the same characteristic polynomial:

All triangular matrices with a common diagonal have the same 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 2024).

#### Text

Wolfram Research (2003), CharacteristicPolynomial, Wolfram Language function, https://reference.wolfram.com/language/ref/CharacteristicPolynomial.html (updated 2024).

#### CMS

Wolfram Language. 2003. "CharacteristicPolynomial." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2024. 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

#### BibTeX

@misc{reference.wolfram_2024_characteristicpolynomial, author="Wolfram Research", title="{CharacteristicPolynomial}", year="2024", howpublished="\url{https://reference.wolfram.com/language/ref/CharacteristicPolynomial.html}", note=[Accessed: 13-August-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_characteristicpolynomial, organization={Wolfram Research}, title={CharacteristicPolynomial}, year={2024}, url={https://reference.wolfram.com/language/ref/CharacteristicPolynomial.html}, note=[Accessed: 13-August-2024 ]}