Eigenvalues
Eigenvalues of a 2×2 or 3×3 matrix from the characteristic polynomial, with eigenvectors for the 2×2 case.
trace 7 · determinant 10
An eigenvector is a direction a matrix does not rotate — it only stretches it, by a factor called the eigenvalue. Find them from det(A − λI) = 0. For a 2×2 this is λ² − (trace)λ + (determinant) = 0, so [[4,1],[2,3]] gives λ = 2 and 5.
The directions a matrix leaves alone
A matrix usually does two things to a vector: turns it and changes its length. For most inputs both happen at once, which makes the transformation hard to picture. Eigenvectors are the exceptional directions where only the stretching happens — the vector comes out pointing the same way (or exactly backwards), scaled by λ.
That is why they matter well beyond linear algebra courses. Principal component analysis finds the eigenvectors of a covariance matrix; a structure’s resonant frequencies are the eigenvalues of its stiffness and mass matrices; and the long-run behaviour of a Markov chain is governed by the eigenvector of its transition matrix with eigenvalue 1.
Two checks that catch most errors
The eigenvalues of any square matrix sum to its trace and multiply to its determinant. Both fall out of the characteristic polynomial, and both are quick to verify by hand — so if your eigenvalues do not add to the sum of the diagonal, something went wrong before you got to the roots. The tool shows both checks alongside the answer for exactly that reason.
the characteristic polynomial; its roots are the eigenvalues
- 1 Subtract λ from the diagonal. For [[4,1],[2,3]] that gives the matrix [[4−λ, 1], [2, 3−λ]].
- 2 Take the determinant and set it to zero. (4−λ)(3−λ) − 1×2 = λ² − 7λ + 10 = 0.
- 3 Check the shortcut. The trace is 4 + 3 = 7 and the determinant is 12 − 2 = 10, which matches the polynomial directly.
- 4 Solve for λ. Factoring gives (λ − 2)(λ − 5) = 0, so the eigenvalues are 2 and 5.
- 5 Verify against trace and determinant. 2 + 5 = 7 and 2 × 5 = 10, so both checks hold.
What the eigenvalues tell you
Read off the spectrum before doing anything else with the matrix.
| Spectrum | What it means |
|---|---|
| A zero eigenvalue | The matrix is singular — determinant 0, not invertible |
| All positive and real | Positive definite, if the matrix is symmetric |
| A complex conjugate pair | The transformation rotates; no real direction is preserved |
| Repeated eigenvalue | May have fewer independent eigenvectors than its multiplicity |
| All |λ| < 1 | Repeated application shrinks every vector towards zero |
| An eigenvalue of exactly 1 | That direction is fixed — the steady state of a Markov chain |
What the numbers here are, and are not
The characteristic polynomial’s coefficients are computed exactly from the entries — trace, the sum of the principal minors and the determinant are just arithmetic. Only the root-finding is numerical, and a root that is very close to a whole number is confirmed against the polynomial before being shown as one, so an eigenvalue of 2 − √2 stays irrational rather than being rounded into something tidier.
Two limitations worth naming. Repeated roots are inherently ill-conditioned — a tiny change in the matrix can move them noticeably — which is a property of the problem rather than of any particular method. And eigenvectors are shown for the 2×2 case with distinct real eigenvalues, where they can be written down directly; a defective matrix with fewer eigenvectors than eigenvalues needs generalised eigenvectors, which is beyond what this page computes.