Eigenvalue Calculator for 2×2 and 3×3 Matrices

Type the entries of a square matrix and instantly read off its eigenvalues, determinant, trace and characteristic polynomial — all computed live in your browser, including complex eigenvalues.

How the eigenvalue computation works

An eigenvalue λ of a square matrix A is a scalar for which there exists a non-zero vector v (the eigenvector) satisfying A·v = λ·v. Rearranging gives (A − λI)·v = 0, which has a non-trivial solution only when the matrix A − λI is singular. That condition is captured by the characteristic polynomial det(A − λI) = 0.

For a 2×2 matrix the characteristic polynomial is the quadratic λ² − tr(A)·λ + det(A) = 0, where the trace tr(A) is the sum of the diagonal entries. This calculator solves it with the standard quadratic formula, returning a real pair when the discriminant is non-negative and a complex-conjugate pair otherwise. For a 3×3 matrix the characteristic polynomial is a cubic λ³ − c₂λ² + c₁λ − c₀ = 0, whose coefficients are the trace, the sum of the principal 2×2 minors, and the determinant. The cubic is solved analytically using Cardano's method together with the trigonometric form for three real roots, so symmetric matrices return clean real eigenvalues while rotation-like matrices correctly reveal complex pairs.

Two quick sanity checks are always shown: the eigenvalues must sum to the trace and multiply to the determinant. Eigenvalues underpin principal component analysis, stability analysis of dynamical systems, vibration modes, Markov-chain steady states and Google's PageRank. Because everything runs locally with no network calls, you can paste sensitive data and explore freely.

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