Characteristic Polynomial

Characteristic Polynomial Calculator

Matrix:

Size of the Matrix

Characteristic Polynomial Calculator

The characteristic polynomial calculator computes the characteristic polynomial of a square matrix of any order (2×2, 3×3, 4×4, etc.). It helps you quickly find the eigenvalues and simplifies solving determinant equations for control systems or linear algebra problems.

What is a Characteristic Polynomial?

The characteristic polynomial of a square matrix is a polynomial whose roots are the eigenvalues of the matrix. It is defined as:

f(λ) = det(A − λI) = 0

Where:

A is the square matrix.
λI is the identity matrix multiplied by λ.

The characteristic polynomial is obtained by calculating the determinant of (A − λI). Solving this polynomial gives the eigenvalues of the matrix.

Example: Characteristic Polynomial of a 3×3 Matrix

Find the characteristic polynomial of the matrix:

\[ A = \begin{bmatrix} 3 & 5 & 2 \\ 7 & 9 & 3 \\ 4 & 5 & 4 \end{bmatrix} \]

Step 1: Subtract λ from the diagonal elements of the matrix:

\[ A - \lambda I = \begin{bmatrix} 3-\lambda & 5 & 2 \\ 7 & 9-\lambda & 3 \\ 4 & 5 & 4-\lambda \end{bmatrix} \]

Step 2: Compute the determinant of (A − λI) to get the characteristic polynomial:

\[ \text{Characteristic Polynomial} = -\lambda^3 + 16\lambda^2 - 17\lambda - 19 \]

This polynomial can then be solved to find the eigenvalues of the matrix.

How the Characteristic Polynomial Calculator Works

Input:

Enter the size of the square matrix.
Enter all coefficient values of the matrix.
Click the Calculate button.

Output:

The characteristic polynomial equation.
Optionally, the roots (eigenvalues) of the polynomial.

Applications

Solving linear differential equations.
Determining eigenvalues for control systems.
Matrix analysis in linear algebra.

Characteristic Polynomial Calculator

Characteristic Polynomial Calculator

What is a Characteristic Polynomial?

Example: Characteristic Polynomial of a 3×3 Matrix

How the Characteristic Polynomial Calculator Works

Input:

Output:

Applications

References