What are eigenvalues?

An eigenvalue λ of a matrix A is a scalar such that Av = λv for some non-zero vector v. Geometrically, the matrix only stretches or flips the eigenvector v without changing its direction.

How is the characteristic polynomial formed?

Subtract λ times the identity matrix from A to get (A - λI), then take the determinant. Setting det(A - λI) = 0 gives the characteristic polynomial whose roots are the eigenvalues.

Can eigenvalues be complex numbers?

Yes. Real non-symmetric matrices can have complex conjugate eigenvalue pairs. This calculator displays them in a + bi form when they arise from the quadratic formula.

What if two eigenvalues are equal (repeated)?

A repeated eigenvalue is called a defective or degenerate eigenvalue. The calculator detects this and still returns a valid eigenvector for the eigenspace, though the eigenspace may have dimension less than the algebraic multiplicity.

Are the eigenvectors normalized?

Yes. This tool normalizes each eigenvector to unit length (magnitude 1) for consistency. The direction is what matters mathematically; any scalar multiple is also a valid eigenvector.

Does this work for non-square or larger matrices?

Eigenvalues are only defined for square matrices. This calculator supports 2x2 and 3x3. For larger matrices, specialized numerical software (MATLAB, NumPy, Wolfram Alpha) is recommended.

Eigenvalue Calculator

Enter a 2x2 or 3x3 matrix to instantly compute its eigenvalues and eigenvectors with full step-by-step output.

Matrix Input

Enter a matrix and click Calculate to see eigenvalues and eigenvectors.

Summary

This eigenvalue calculator finds all eigenvalues and corresponding eigenvectors for any real 2x2 or 3x3 matrix. It derives the characteristic polynomial det(A - λI) = 0, solves for λ, and computes a basis eigenvector for each eigenvalue. Results include the full characteristic equation and normalized eigenvectors formatted for easy reading.

How it works

Select the matrix size: 2x2 or 3x3.
Enter the matrix entries in the grid (decimal values are supported).
Click "Calculate" to run the computation.
The tool builds the characteristic polynomial det(A - λI) and solves for all eigenvalues λ.
For each eigenvalue, it solves (A - λI)v = 0 to find a corresponding eigenvector.
Results appear instantly with the characteristic equation, eigenvalues, and eigenvectors.

Use cases

Verify hand-computed eigenvalues for linear algebra homework.
Analyze the stability of a dynamic system by checking eigenvalue signs.
Find principal axes for a 2D or 3D stress tensor in engineering.
Check whether a matrix is diagonalizable before further computation.
Understand the principal components in a small covariance matrix.
Confirm that eigenvectors are orthogonal for symmetric matrices.
Prototype numerical methods before implementing them in code.
Explore how eigenvalues change as matrix entries change.

Frequently Asked Questions

Last updated: 2026-05-29 · Reviewed by Nham Vu

Eigenvalue Calculator

Matrix Input

Characteristic Polynomial

Eigenvalues

Eigenvectors (unit length)

Summary

How it works

Use cases

Frequently Asked Questions

Eigenvalue Calculator

Use the Eigenvalue Calculator

Matrix Input

Characteristic Polynomial

Eigenvalues

Eigenvectors (unit length)

Summary

How it works

Use cases

Frequently Asked Questions