Eigenvalues are the numbers λ for which A·v = λ·v has a non-zero solution v, the eigenvector. They are the roots of the characteristic polynomial det(A − λI) = 0. The eigenvalues of [[2,0],[0,3]] are 2 and 3, their sum equals the trace, and their product equals the determinant.
What are eigenvalues and eigenvectors?
Most vectors change direction when a matrix is applied. Eigenvectors are the special ones that only get stretched or flipped, by a factor called the eigenvalue. If A·v = λ·v, then v is an eigenvector with eigenvalue λ. Geometrically, a shear or stretch transformation has directions that it leaves on the same line, and the eigenvalues tell you by how much it scales along them. This is useful to understand stability of systems, vibration modes, principal components in statistics and the long-run behaviour of Markov chains.
How to calculate them
- Form the characteristic equation det(A − λI) = 0. For a 2×2 matrix this is λ² − (trace)λ + det = 0.
- Solve the polynomial for λ. Degree 2 can be solved with the quadratic formula (see the Quadratic Formula Calculator), while degree 3 and 4 are solved numerically.
- For each eigenvalue, solve (A − λI)v = 0 to find the eigenvector.
The calculator builds the characteristic polynomial with the Faddeev-LeVerrier method and finds all roots numerically, including complex ones, then solves for the eigenvector of each real eigenvalue and scales it so its largest entry is 1.
Quick checks that always work
- The sum of the eigenvalues equals the trace of the matrix.
- The product of the eigenvalues equals the determinant (see the determinant calculator).
- A triangular matrix has its diagonal entries as eigenvalues.
- A symmetric real matrix has only real eigenvalues.
- A matrix is invertible exactly when none of its eigenvalues are zero.
Complex eigenvalues
A real matrix can have complex eigenvalues, which come in conjugate pairs. A pure rotation matrix [[0,−1],[1,0]] has eigenvalues +i and −i, because no real direction stays fixed. The calculator shows these as a + bi and marks the eigenvector as complex.
For other matrix operations, see the matrix calculator and the inverse matrix calculator.
Worked example: a 2×2 matrix by hand
For A = [[4,1],[2,3]] the trace is 7 and the determinant is 10, so the characteristic equation is λ² − 7λ + 10 = 0. This factorises as (λ − 5)(λ − 2) = 0, giving eigenvalues 5 and 2. For λ = 5, solve (A − 5I)v = 0: the matrix [[−1,1],[2,−2]] gives v = (1, 1). For λ = 2 the matrix [[2,1],[2,1]] gives v = (1, −2). Enter this matrix as a 2×2 and you will see the same numbers, each eigenvector scaled so the largest entry is 1, so (1, −2) shows as (−0.5, 1).
Eigenvalues also describe long-term behaviour. If you repeatedly multiply a vector by A, the component along the eigenvector with the largest absolute eigenvalue (here 5) dominates. That is the idea behind Google’s original PageRank and population models such as the Leslie matrix.
How to use this calculator
- Choose the matrix size (2×2, 3×3 or 4×4) and the operation.
- Type your numbers into the matrix grids. Decimals and negatives are fine.
- The result updates instantly, including checks such as A × A⁻¹.
- Change the operation to explore the same matrix in other ways.
Key figures at a glance
| Sum of eigenvalues | = trace |
|---|---|
| Product of eigenvalues | = determinant |
| Equation | det(A − λI) = 0 |
| Sizes | 2×2 to 4×4 |
Frequently asked questions
Solve det(A − λI) = 0. For a 2×2 matrix this is the quadratic λ² − trace·λ + determinant = 0.
A non-zero vector that is only scaled by the matrix: A·v = λ·v.
Yes, for real non-symmetric matrices they can come in complex conjugate pairs.
Three, counted with multiplicity, though some may repeat or be complex.
Any multiple of an eigenvector is also an eigenvector. We scale so the largest entry is 1 for easy comparison.