2×2 Matrix Eigenvalue and Eigenvector Calculator
Calculate eigenvalues and eigenvectors of a 2×2 matrix.
Find the characteristic polynomial, determinant, trace, and diagonalizability with step-by-step solution.
Eigenvalues and Eigenvectors For a matrix A, an eigenvector v and eigenvalue λ satisfy: Av = λv The eigenvector’s direction is unchanged by the transformation — only its length scales by λ. Named from German “eigen” = own/characteristic. Introduced by David Hilbert (Germany, 1904).
For a 2×2 Matrix A = [[a, b], [c, d]] Characteristic polynomial: det(A − λI) = 0 det([[a−λ, b], [c, d−λ]]) = (a−λ)(d−λ) − bc = 0 λ² − (a+d)λ + (ad−bc) = 0 λ² − tr(A)λ + det(A) = 0
Solving for Eigenvalues Using the quadratic formula: λ = [tr(A) ± √(tr(A)² − 4det(A))] / 2 Discriminant Δ = tr(A)² − 4det(A) Δ > 0: two distinct real eigenvalues Δ = 0: one repeated real eigenvalue Δ < 0: two complex conjugate eigenvalues (no real eigenvectors)
Finding Eigenvectors For each eigenvalue λ, solve (A − λI)v = 0. For 2×2: (a−λ)v₁ + b·v₂ = 0 → v = [−b, a−λ] (or [d−λ, −c]) Eigenvectors are not unique — any scalar multiple is also an eigenvector.
Physical Meaning Principal stress directions in mechanics (Mohr’s circle). Principal moments of inertia. Vibration modes of structures. Google PageRank (dominant eigenvector of link matrix). Quantum mechanics: observables are eigenvalues of operators.