# 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(*x***I** – **A**), where *x* is the variable of the polynomial and **I** is the corresponding identity matrix.

## Syntax

## Description

#### characteristic_polynomial(Matrix)

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

#### characteristic_polynomial(Matrix, Expression)

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

## 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 |