Eigenvalues and eigenvectors, explained
An eigenvector of a matrix is a direction the matrix only stretches, never turns. The eigenvalue is the stretch factor. They come from det(A − λI) = 0, and they show up in systems of linear ODEs, vibrations and stability.
Finding them
Write A − λI and set its determinant to zero. That gives the characteristic polynomial, and its roots are the eigenvalues. For each eigenvalue, solve (A − λI)v = 0. The system always has a nonzero solution, because the determinant is zero. Any nonzero multiple of that solution is also an eigenvector.
Two checks that catch most mistakes
The eigenvalues add up to the trace, the sum of the diagonal. They multiply to the determinant. Both checks take seconds and catch most arithmetic slips before they spread into the eigenvectors.
Complex eigenvalues
A real matrix can have complex eigenvalues, always in conjugate pairs. Geometrically there is no real direction left unturned: the matrix rotates every vector. In a system of ODEs, complex eigenvalues mean oscillation, and their real part decides whether it grows or decays.
Worked example
For A = [[2, 1], [1, 2]], the trace is 4 and the determinant is 3, so λ² − 4λ + 3 = 0, giving λ = 3 and λ = 1.
For λ = 3, A − 3I = [[−1, 1], [1, −1]], so v = (1, 1). For λ = 1, v = (1, −1). Checks: 3 + 1 = 4, the trace, and 3 × 1 = 3, the determinant. These are the calculator’s default inputs.
2×2 eigenvalue calculator
A = [[a, b], [c, d]]
Common mistakes
- Computing det(A) − λ instead of det(A − λI).
- Treating the eigenvector as unique. Any nonzero multiple works, so (2, 2) is as correct as (1, 1).
- Accepting the zero vector. It satisfies Av = λv for every λ, which is exactly why it is excluded.