Eigenvalue and eigenvector calculator
Find the eigenvalues and eigenvectors of a square matrix, plus its characteristic polynomial, computed with the Jacobi and QR algorithms.
Enter a square matrix, open the Advanced tab and use eig(A) for the eigenvalues or eigvec(A) for the eigenvectors. The Decompositions tab additionally gives the characteristic polynomial in exact arithmetic.
What an eigenvector is
Most vectors change direction when a matrix acts on them. An eigenvector is one of the rare directions that survives: the matrix only stretches or shrinks it. The factor by which it does so is the eigenvalue. Formally, A·v = λ·v for a non-zero vector v.
They come from the characteristic equation det(A − λI) = 0, which is where the determinant reappears. For a 2×2 matrix that equation is a quadratic; solving it gives both eigenvalues.
Worked example
| 2 | 1 |
| 1 | 2 |
The eigenvector for λ = 3 is (1, 1): the matrix triples anything along that diagonal. For λ = 1 it is (1, −1), which the matrix leaves untouched.
A limitation worth knowing
This calculator computes eigenvalues numerically — Jacobi rotations for symmetric matrices, the QR algorithm otherwise. Both are iterative, so eigenvalues are the one operation without a step-by-step solution: the intermediate iterations would teach you nothing. They also converge to real eigenvalues only; a rotation matrix, whose eigenvalues are complex, will report that it did not converge rather than invent an answer.
Common mistakes
- Expecting eigenvalues for a rectangular matrix. They are defined only for square matrices — singular values are the rectangular analogue.
- Treating an eigenvector as unique. Any non-zero multiple of an eigenvector is also an eigenvector; only the direction is determined.
- Assuming they are always real. Real matrices can perfectly well have complex eigenvalues.