Eigenvalue Calculator — Free Online

The Eigenvalue Calculator finds the eigenvalues and eigenvectors of square matrices. Eigenvalues and eigenvectors are fundamental concepts in linear algebra, used in physics, engineering, computer science, and data analysis.

Enter a square matrix (2x2 or 3x3) and the calculator will compute all eigenvalues and their corresponding eigenvectors.

Enter the 2\u00d72 matrix elements:

a\u2081\u2081

a\u2081\u2082

a\u2082\u2081

a\u2082\u2082

\u03bb\u2081

\u03bb\u2082

Trace

Determinant

Your calculations will appear here

How to Use the Eigenvalue Calculator

Select the matrix size (2x2 or 3x3).
Enter the matrix elements.
The calculator finds all eigenvalues and their corresponding eigenvectors.

Formula and Theory

An eigenvalue of a matrix A is a scalar lambda such that Av = lambda*v for some non-zero vector v (the eigenvector). This means the matrix transformation only scales the vector without changing its direction.

Eigenvalues are found by solving the characteristic equation det(A - lambda*I) = 0. For a 2x2 matrix, this gives a quadratic equation; for 3x3, a cubic equation.

Worked Examples

Example 1: 2x2 eigenvalues

Problem: Find the eigenvalues of [[4, 1], [2, 3]]

Solution: Characteristic equation: (4-lambda)(3-lambda) - 2 = lambda^2 - 7*lambda + 10 = 0

Answer: lambda = 5 and lambda = 2

Frequently Asked Questions

What are eigenvalues used for?

Eigenvalues are used in principal component analysis (PCA), quantum mechanics, vibration analysis, stability analysis, Google's PageRank algorithm, and many other applications.

Can eigenvalues be complex numbers?

Yes. A real matrix can have complex eigenvalues, which always come in conjugate pairs (a + bi and a - bi).

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