Characteristic Polynomial Calculator
Find the characteristic polynomial of square matrices.
Table of Contents
How to Use
- Select the matrix size (2×2 or 3×3).
- Enter the numerical values for each matrix entry.
- Click Calculate to compute the characteristic polynomial.
- View the polynomial equation, determinant, and trace.
What Is the Characteristic Polynomial?
The characteristic polynomial of a square matrix A is defined as p(λ) = det(A - λI), where λ is a variable, I is the identity matrix, and det denotes the determinant.
The roots of this polynomial are the eigenvalues of the matrix, making it fundamental in linear algebra for understanding matrix properties and transformations.
Key Properties
- The degree of the polynomial equals the matrix size (n×n matrix gives degree n polynomial)
- The constant term is the determinant of the matrix
- The coefficient of λⁿ⁻¹ is the negative trace of the matrix
- The roots of the polynomial are the eigenvalues
Frequently Asked Questions
- What is the characteristic polynomial used for?
- The characteristic polynomial is used to find eigenvalues of a matrix by solving p(λ) = 0. Eigenvalues are essential in many applications including stability analysis, principal component analysis, and quantum mechanics.
- How do I find eigenvalues from the characteristic polynomial?
- Solve the polynomial equation p(λ) = 0. The solutions (roots) are the eigenvalues of the matrix.