Characteristic polynomial

The characteristic polynomial of a square matrix is a polynomial that is invariant under matrix similarity and has the eigenvalues of the matrix as eigenvectors. It is defined as P(x) = det(xI – A), where x is the variable of the polynomial and I is the corresponding identity matrix.

Syntax

characteristic_polynomial(Matrix)
characteristic_polynomial(Matrix, Expression)

Description

Given a matrix, it returns its characteristic polynomial in the variable x1LaTeXx1.

linear_algebra.characteristic_polynomial1.calc.png

Given a matrix and an expression, it returns its characteristic polynomial evaluated in the desired expression.

linear_algebra.characteristic_polynomial2.calc.png

Options

Below is a complete list of options that may be used in the characteristic_polynomial function.

Option

Description

Format

Default value

method

We can choose the method to be used from the following: adjoint_matrix, determinant, hessenberg and hessenberg_householder.

{method=”adjoint_matrix”}

hessenberg_householder

exact_computations

We can choose to perform or not exact computations.

{exact_computations=false}

true, but depends on the input