Eigenvalue Solver
Find the eigenvalues and eigenvectors of a real 2×2, 3×3 or 4×4 matrix, including complex eigenvalues. Shows the characteristic polynomial and checks the result against the trace and determinant.
What are Eigenvalues?
Most vectors change direction when you multiply them by a matrix. An eigenvector v is special: A only stretches or shrinks it, Av = λv. The stretch factor λ is its eigenvalue.
Eigenvalues are the roots of the characteristic polynomial det(A − λI). They describe a system’s natural behaviour: vibration frequencies, stability of differential equations, principal axes in statistics (PCA) and the energy levels of quantum systems.
Transformation View
AEigenvalue Properties
| Matrix | Eigenvalues |
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Solving Methods
For a 2×2 matrix the eigenvalues follow from the quadratic formula using only the trace and the determinant.
For each eigenvalue, solve this homogeneous system by row reduction. Every non-zero solution is an eigenvector; they form the eigenspace of λ.
Repeatedly multiply a starting vector by A and normalise. It converges to the eigenvector of the largest |λ| — the idea behind Google’s PageRank.
Sources & Verification
- G. Strang, Introduction to Linear Algebra, 5th ed., Wellesley-Cambridge Press, 2016 — Ch. 6 (Eigenvalues and Eigenvectors).
- D. C. Lay, S. R. Lay & J. J. McDonald, Linear Algebra and Its Applications, 5th ed., Pearson, 2016 — Ch. 5 (Eigenvalues and Eigenvectors).