Linear algebra
Eigenvalue and eigenvector calculator
Enter a square matrix and get the characteristic polynomial, the eigenvalues — real or complex — and the associated eigenvectors.
What eigenvalues and eigenvectors are
An eigenvector of a matrix A is a non-zero vector that the transformation merely stretches or compresses without changing its direction: A v = λ v. The number λ is the associated eigenvalue and says by how much that vector is scaled. A negative eigenvalue means the direction is also flipped; one equal to 1 means the vector stays exactly where it is.
Eigenvalues are found by setting the determinant of A − λI to zero: this gives the characteristic polynomial, of degree equal to the order of the matrix, and its roots are the eigenvalues. An n × n matrix always has n eigenvalues counted with multiplicity, but they can be complex even when the matrix is made entirely of real numbers: a 90-degree rotation in the plane, for instance, leaves no real direction unchanged and has eigenvalues ±i.
Once an eigenvalue λ is known, its eigenvectors are the non-zero solutions of the homogeneous system (A − λI) v = 0, that is the null space of that matrix. Because the system is homogeneous, if v is an eigenvector so is any multiple of it: the calculator normalises each vector so that its largest component is 1, so two runs always return the same representation.
Two quick checks let you verify the result by hand: the sum of the eigenvalues equals the trace of the matrix, that is the sum of the entries on the main diagonal, and their product equals the determinant. For the matrix [[4, 1], [2, 3]] the eigenvalues are 5 and 2: their sum is 7, which is 4 + 3, and their product is 10, which is the determinant.
Common mistakes
- Accepting the zero vector as an eigenvector: it satisfies the equation for any λ, which is precisely why it is excluded by definition.
- Assuming the eigenvalues of a real matrix must be real: that is guaranteed only for symmetric matrices, not in general.
- Confusing algebraic and geometric multiplicity: a repeated eigenvalue does not necessarily have two independent eigenvectors, and when it does not, the matrix is not diagonalisable.
- Treating two eigenvectors as different because they are written at different scales: (1, 2) and (2, 4) span the same direction and the same eigenspace.
- Looking for the eigenvalues of a non-square matrix: the definition does not apply, and singular values are what that case calls for.
Frequently asked questions
How do you calculate the eigenvalues of a matrix?
Solve det(A − λI) = 0, that is set the characteristic polynomial to zero. For a 2 × 2 matrix this gives λ² − (trace)λ + determinant = 0, a quadratic equation; at higher orders a numerical method is needed.
Can a real matrix have complex eigenvalues?
Yes, and they always come in conjugate pairs. Rotation matrices are the standard case: a 90-degree rotation in the plane leaves no real direction unchanged, and its eigenvalues are i and −i. Symmetric matrices are the exception: their eigenvalues are always real.
Why is my eigenvector different from the one in the textbook?
An eigenvector is only defined up to a scalar factor: if v is an eigenvector, so are 2v and −v. This page normalises the largest component to 1, while other texts normalise the length to 1 or keep the entries as whole numbers.
How are eigenvalues, trace and determinant related?
The sum of the eigenvalues equals the trace of the matrix and their product equals the determinant. These are two quick, reliable checks: if they do not match, there is an error in the calculation or in the data entered.
When is a matrix diagonalisable?
When it has as many independent eigenvectors as its order, that is when the geometric multiplicity of every eigenvalue matches its algebraic multiplicity. Real symmetric matrices always are, and with eigenvectors orthogonal to one another.
How this calculation works
The characteristic polynomial p(λ) = det(A − λI) is built with the Faddeev-LeVerrier method, which derives the coefficients from the traces of successive products without any symbolic expansion. Its roots — the eigenvalues, real or complex — are found with the Durand-Kerner method, which approximates all roots of a monic polynomial at once. For each real eigenvalue λ the eigenvectors are a basis of the null space of (A − λI), computed by elimination with a tolerance suited to the fact that λ comes from an iterative method.
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