Eigenvalue Calculator

This eigenvalue calculator finds the eigenvalues of a 2×2 or 3×3 matrix — the roots of its characteristic equation — with the trace and determinant alongside.

Eigenvalues (λ) —
Trace —
Determinant —

How to Find Eigenvalues

Enter a 2×2 or 3×3 matrix, one row per line.

1

Enter the matrix

Type each row on its own line, with entries separated by spaces or commas. The matrix must be square — 2×2 or 3×3.

2

Read the eigenvalues

The eigenvalues are the roots of the characteristic equation det(A − λI) = 0, shown largest first.

3

Check with trace and determinant

The eigenvalues sum to the trace and multiply to the determinant — a quick way to sanity-check the result.

Frequently Asked Questions

What is an eigenvalue?
An eigenvalue λ of a square matrix A is a number for which Av = λv has a non-zero solution vector v, called an eigenvector. Geometrically, A stretches that eigenvector by the factor λ without changing the direction it points along. An n×n matrix has n eigenvalues counted with multiplicity, and their sum equals the trace while their product equals the determinant. For [[2, 0], [0, 3]] the eigenvalues are simply 2 and 3.
How are eigenvalues calculated?
Eigenvalues are the roots of the characteristic equation det(A − λI) = 0. For a 2×2 matrix that expands to a quadratic, λ² − (trace)λ + (determinant) = 0; for a 3×3 it is a cubic. For [[4, 1], [2, 3]] the trace is 7 and the determinant is 10, giving λ² − 7λ + 10 = 0 and eigenvalues 5 and 2. This calculator forms and solves that polynomial and returns the real eigenvalues.
Can eigenvalues be complex?
Complex eigenvalues are common even for matrices with entirely real entries, and for a real matrix they always come in conjugate pairs. The rotation matrix [[0, −1], [1, 0]] has characteristic equation λ² + 1 = 0, so its eigenvalues are the imaginary pair i and −i and no real eigenvector exists. A real symmetric matrix, by contrast, is guaranteed to have only real eigenvalues. This calculator reports the real eigenvalues and notes when a complex pair is present.
Is my data sent anywhere?
Your figures stay on your device when you use the eigenvalue calculator. The eigenvalue calculator works in the browser using JavaScript, with no server round-trip at any point, so your matrix entries are never transmitted. Nothing is logged, stored or shared with a third party. That also means the tool keeps working offline.

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