Eigenvalue and Eigenvector Calculator

Eigenvalues are found by solving the characteristic equation det(A − λI) = 0, where A is your matrix, λ is the eigenvalue, and I is the identity matrix. For a 2×2 matrix this yields a quadratic equation; for a 3×3 matrix it yields a cubic. The roots of that polynomial are the eigenvalues.

Once you have an eigenvalue λ, substitute it back into (A − λI)v = 0 and solve the resulting system of linear equations for the vector v. The non-zero solutions are the eigenvectors corresponding to that eigenvalue. Each eigenvalue has at least one associated eigenvector.

An n×n matrix has exactly n eigenvalues when counted with multiplicity (including complex ones). So a 2×2 matrix has 2 eigenvalues and a 3×3 matrix has 3 eigenvalues. Some of these may be repeated (algebraic multiplicity > 1) or complex numbers.

For a 3×3 matrix, expand the characteristic polynomial det(A − λI) = 0 to get a cubic equation −λ³ + tr(A)λ² − … + det(A) = 0. Solve this cubic for λ using methods like factoring, numerical approximation, or Cardano's formula. This calculator handles it numerically for you.

Eigenvectors corresponding to distinct eigenvalues of a symmetric (or Hermitian) matrix are always orthogonal to each other. However, for general non-symmetric matrices, eigenvectors from different eigenvalues are not guaranteed to be orthogonal.

Yes, 0 is a valid eigenvalue. If λ = 0 is an eigenvalue of a matrix A, it means the matrix is singular (non-invertible) and its determinant equals zero. The corresponding eigenvectors lie in the null space of A.

For any square matrix, the sum of all eigenvalues equals the trace (sum of diagonal elements), and the product of all eigenvalues equals the determinant. These relationships provide quick sanity checks for your eigenvalue computations.

When the characteristic polynomial has no real roots, eigenvalues appear as complex conjugate pairs (a ± bi). This typically occurs when a matrix represents a rotation or spiral transformation. The corresponding eigenvectors also have complex components, but the real-world interpretation often involves oscillatory behavior.