Eigenvalue Calculator
Find eigenvalues of 2×2 and 3×3 matrices with one click.
Table of Contents
How to Use
- Choose whether you are working with a 2×2 or 3×3 matrix.
- Enter the numeric value for each matrix entry in the grid.
- Select Calculate to evaluate the characteristic polynomial and eigenvalues.
- Review the list of eigenvalues along with the trace, determinant, and notes on complex pairs.
What Is an Eigenvalue?
Given a square matrix A, an eigenvalue λ satisfies A v = λ v for some non‑zero vector v. Solving det(A − λI) = 0 produces the eigenvalues, which describe how the matrix stretches or flips vectors along preferred directions.
The eigenvalues summarize important behavior such as whether repeated multiplication amplifies or damps signals, and whether the transformation rotates or reflects space.
Interpreting the Results
- The trace equals the sum of all eigenvalues.
- The determinant equals the product of all eigenvalues.
- A negative discriminant signals a complex conjugate pair for 2×2 matrices.
- Repeated eigenvalues may indicate special structure such as defective or symmetric matrices.
Frequently Asked Questions
- Why do I get complex eigenvalues?
- Complex eigenvalues appear whenever the characteristic polynomial has negative discriminant or no real roots. They come in conjugate pairs and indicate that the matrix combines rotation with scaling.
- Can I find eigenvectors with this tool?
- This calculator focuses on eigenvalues and invariants. Once you know an eigenvalue, substitute it into (A − λI)v = 0 and solve the resulting linear system to obtain the associated eigenvectors.