Find eigenvalues and eigenvectors for 2×2 matrices exactly, and for symmetric 3×3 matrices by Jacobi iteration.
Result
How to use this tool
Pick 2×2, or symmetric 3×3.
Enter each entry, using 0 where needed.
Read eigenvalues largest first.
Read the matching normalised eigenvectors.
Formula & worked example
[[2,1],[1,2]] has eigenvalues 3 and 1; the symmetric 3×3 matrix with 2s on the diagonal and 1s beside it has eigenvalues about 3.4142, 2 and 0.5858.
Limitations — read this before you rely on it
Real eigenvalues only; 3×3 mode requires a symmetric matrix, and complex 2×2 cases are flagged, not solved.
Frequently asked questions
A scalar λ such that A·v = λ·v for some non-zero vector v, the eigenvector.
Exactly, from the characteristic equation λ² – trace·λ + determinant = 0.
A rotation method that repeatedly zeroes off-diagonal entries of a symmetric matrix until the diagonal holds the eigenvalues.
Some real matrices have complex eigenvalue pairs; this tool covers real eigenvalues and eigenvectors only.
Jacobi iteration is guaranteed for symmetric matrices; general 3×3 eigenproblems need a different solver.