An eigenvector is a direction that a matrix does not rotate: applying A to it only stretches or shrinks it, by a factor called the eigenvalue. Press eigvec(A) and each eigenvector comes back as a column of the result.
What the vectors mean
Formally, v is an eigenvector of A when A·v = λ·v for some scalar λ. Everything a matrix does to space is a mixture of stretching along these special directions, which is why they turn up wherever a linear transformation needs to be understood rather than merely applied — principal component analysis, vibration modes, stability of a differential equation, the PageRank vector.
Eigenvectors are never unique. If v works then so does 2v, and −v, and any non-zero multiple: the equation only constrains the direction. Calculators therefore return a representative, usually normalised to unit length, and yours may differ from a textbook’s by a scale factor while being equally correct. The zero vector is excluded by definition — it would satisfy the equation for every λ and tell you nothing.
Worked example
Take the matrix below, whose eigenvalues are 3 and 1:
| 2 | 1 |
| 1 | 2 |
| 1 |
| 1 |
| 1 |
| 1 |
The direction (1, 1) survives untouched and is merely tripled. The second eigenvector, (1, −1), is scaled by 1 — left exactly as it was. The two are perpendicular, which is no accident: this matrix is symmetric, and symmetric matrices always have orthogonal eigenvectors and real eigenvalues.
Common mistakes
- Expecting a unique answer. Any non-zero multiple of an eigenvector is an eigenvector. Compare directions, not entries.
- Accepting the zero vector. It is excluded by definition, however well it satisfies the equation.
- Assuming n independent eigenvectors always exist. A defective matrix has a repeated eigenvalue with too few eigenvectors, and cannot be diagonalised.
- Forgetting complex results. A rotation matrix has no real eigenvector at all — no real direction survives a rotation — and its eigenvalues are complex.