An eigenvector is a direction the matrix does not turn. It only stretches it, and the eigenvalue is the stretch factor. They are the roots of det(A − λI) = 0, and they must sum to the trace and multiply to the determinant — two checks that take seconds.
What an eigenvector is
Most directions get turned by a matrix. A few do not — they come out pointing the same way, only longer or shorter. Those are the eigenvectors, and the scaling factor is the eigenvalue.
Av = λv. The matrix, applied to that particular vector, behaves like a single number.
That is what makes them useful. A matrix is complicated; a number is not. If a vector can be written in terms of eigenvectors, applying the matrix a hundred times is a hundredth power of each eigenvalue rather than a hundred matrix multiplications.
Geometrically they are the axes of the transformation. Stretching a rubber sheet unevenly, the eigenvectors are the directions that stay straight.
The characteristic polynomial
Av = λv rearranges to (A − λI)v = 0, and that has a non-zero solution exactly when A − λI is singular — which means its determinant is zero.
So det(A − λI) = 0 is the equation, and expanded it is a polynomial of degree n in λ. Its roots are the eigenvalues, and there are n of them counting repeats.
This page computes that polynomial exactly, by the Faddeev–LeVerrier recurrence, which builds the coefficients from traces of successive powers and never leaves the rationals.
The roots are a different matter. A polynomial of degree five or more generally has no root formula at all, and even a quadratic usually gives irrational answers. So the eigenvalues here are decimal, and the page says which half of the answer is exact rather than presenting both the same way.
When they are complex
A real matrix can have complex eigenvalues, and it is not a pathology.
Take a 90° rotation. Every direction gets turned, so there is no real direction it leaves unturned — and that is exactly what having no real eigenvalues means. The complex ones encode the rotation angle instead.
For a real matrix they always come in conjugate pairs, so the count of complex eigenvalues is always even. An odd-sized real matrix therefore always has at least one real eigenvalue, which is why every rotation in three dimensions has an axis.
Symmetric matrices are the reliable case. A real symmetric matrix always has all real eigenvalues, and eigenvectors that are perpendicular to each other. That guarantee is why symmetric matrices turn up everywhere they can be arranged for — covariance matrices, stiffness matrices, graph Laplacians.
Two checks worth doing
Two identities constrain the answer, and both take seconds.
The eigenvalues sum to the trace — the sum of the diagonal.
The eigenvalues multiply to the determinant.
Both follow from the characteristic polynomial: its coefficients are, up to sign, the elementary symmetric functions of the roots, and the trace and determinant are the first and last of those.
For a 2×2 they pin the answer down completely, since two numbers with a known sum and product are determined. This page reports both alongside the computed values so the check is visible rather than left as an exercise.
A zero eigenvalue is the same statement as a zero determinant: the matrix is singular, and there is a direction it sends to nothing.
Where it gets used
Stability. A system settles when every eigenvalue has magnitude below one, and runs away when any exceeds it. That single test decides whether a control system holds.
Principal component analysis. The eigenvectors of a covariance matrix are the directions of greatest variation, and the eigenvalues say how much each accounts for. It is the standard way to reduce dimensions.
Vibration and resonance. The eigenvalues of a structure’s stiffness matrix are its natural frequencies — which is why bridges are analysed this way before they are built.
PageRank. The original web ranking algorithm finds the dominant eigenvector of a link matrix. Quantum mechanics does the same thing under a different name: energy levels are eigenvalues.
Sources and methodology
Eigenvalues and the algorithms for them are standard; these are the references.
Method. The characteristic polynomial is computed by the Faddeev–LeVerrier recurrence on exact rationals, which keeps it exact — unlike expanding det(A − λI) symbolically, and unlike anything that starts from numerically computed eigenvalues. The eigenvalues are the roots of that polynomial and are generally irrational, so those are decimal, and the page labels the two differently rather than presenting both as equally exact. The suite verifies the polynomial through Cayley–Hamilton: substituting the matrix into its own characteristic polynomial must give exactly the zero matrix. That engine is verified on every change against 104 hand-written assertions, including that every generated matrix satisfies its own characteristic polynomial exactly — the Cayley–Hamilton theorem — and that the real eigenvalues sum to the trace and multiply to the determinant. The count and the per-case breakdown are published on the formula verification page.
Read the guide
The determinant the eigenvalues have to multiply to is computed exactly on the Determinant Calculator.