2×2 Matrix Eigenvalue and Eigenvector Calculator

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 a factor of λ. The name comes from the German “eigen”, meaning own or characteristic. David Hilbert (Germany) coined the term Eigenwert in 1904, but the idea is far older: Euler was studying the principal axes of rotating bodies in the 1750s, and Cauchy proved in 1829 that a symmetric matrix always has real eigenvalues. Hilbert named the thing; he did not invent it.

A note on the zero vector. Every matrix satisfies A·0 = λ·0 for any λ at all, so the zero vector is deliberately excluded from the definition. An eigenvector must be non-zero, otherwise eigenvalues would carry no information. Any answer of [0, 0] is a bug, not a result.

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.