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. All three are different views of the same question, which the rest of this page works through.
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.
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, the stability of a differential equation, the PageRank vector.
The characteristic polynomial
The defining equation A·v = λ·v rearranges to (A − λI)·v = 0. A non-zero v can only solve that if A − λI is singular — otherwise the only solution is v = 0 — and a matrix is singular exactly when its determinant vanishes. So the eigenvalues are the values of λ making det(A − λI) = 0, and that expression is the characteristic polynomial.
For an n×n matrix it has degree n, and two of its coefficients can be read straight off the matrix without expanding anything: the coefficient just below the leading term is the negative of the trace, and the constant term is the determinant up to a sign of (−1)ⁿ. Both make excellent checks on a hand computation.
Worked example
| 2 | 1 |
| 1 | 2 |
| 2 − λ | 1 |
| 1 | 2 − λ |
The roots 1 and 3 are the eigenvalues. Note the checks: the trace is 4 and the λ coefficient is −4; the determinant is 3 and so is the constant term. The eigenvector for λ = 3 is (1, 1) — the matrix triples anything along that diagonal — and for λ = 1 it is (1, −1), which the matrix leaves untouched. The two are perpendicular, which is no accident: this matrix is symmetric, and symmetric matrices always have orthogonal eigenvectors and real eigenvalues.
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. Compare directions, not entries. The zero vector is excluded by definition — it would satisfy the equation for every λ and tell you nothing.
When there are not enough eigenvectors
An n×n matrix has exactly n eigenvalues counted with multiplicity, but a repeated eigenvalue does not guarantee a matching number of independent eigenvectors. A matrix that comes up short is called defective, and it cannot be diagonalised: there is no basis of eigenvectors to diagonalise it in. Reading algebraic multiplicity as a count of eigenvectors is one of the easiest mistakes to make here.
Real matrices can also have complex eigenvalues. A rotation in the plane has no real eigenvector at all — no real direction survives a rotation — and its eigenvalues are a complex conjugate pair.
How this calculator computes them
Eigenvalues are computed numerically: Jacobi rotations for symmetric matrices, and the QR algorithm with a Wilkinson shift and deflation otherwise. The characteristic polynomial takes a different route — the Faddeev–LeVerrier recurrence, which builds the coefficients from traces of successive powers and stays in exact arithmetic instead of expanding a symbolic determinant.
Because the eigenvalue algorithms are iterative rather than algebraic, eigenvalues are the one operation here without a step-by-step solution: the intermediate iterations are rounding artefacts, not reasoning you could reproduce by hand. They also converge to real eigenvalues only, so a matrix with a complex spectrum reports that it did not converge rather than inventing an answer. Going the other way — finding eigenvalues from the polynomial — is fine up to 3×3 by hand but a bad idea numerically, because polynomial roots are extremely sensitive to error in the coefficients.
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.
- Subtracting λ from every entry. λI is λ on the diagonal and zero elsewhere, so only the diagonal changes.
- Reading multiplicity as a count of eigenvectors. A double root need not bring two independent eigenvectors — when it does not, the matrix is defective.
- Losing the sign convention. det(A − λI) and det(λI − A) differ by (−1)ⁿ. Both are used; pick one and stay with it.