Eigenvalue and Eigenvector Calculator for 2×2 and 3×3 Matrices
Type a 2×2 or 3×3 matrix one row per line. You get the characteristic polynomial, the eigenvalues (exact where possible) and an eigenvector for each real eigenvalue.
- Nothing you type is stored
- Free
- No signup
Eigenvalues
λ = 5, 2
- Trace (sum of λ)
- 7
- Determinant (product of λ)
- 10
Check: the eigenvalues add up to 7 (trace) and multiply to 10 (determinant).
Show working
- A = [4, 1; 2, 3]
- det(A − λI) = 0
- λ² − (tr A)λ + det A = 0
- tr A = 7, det A = 10
- λ² − 7λ + 10 = 0
- λ = (tr A ± √(tr² − 4 det)) / 2 = 5, 2
- λ₁ = 5: (A − 5I)v = 0 → v = (1, 1)
- λ₂ = 2: (A − 2I)v = 0 → v = (−1/2, 1)
Eigenvectors
| λ | Eigenvalue | Eigenvector |
|---|---|---|
| λ₁ | 5 | (1, 1) |
| λ₂ | 2 | (−1/2, 1) |
How to use
- Type the matrix, e.g. “4 1” then “2 3”.
- Read the eigenvalues; repeated ones are marked (×2).
- Check the eigenvector table and open “Show working”.
Good to know
Eigenvalues solve det(A − λI) = 0. For 2×2 this is λ² − (tr A)λ + det A = 0, solved exactly with surds. For 3×3 it is λ³ − (tr A)λ² + (M₁₁ + M₂₂ + M₃₃)λ − det A = 0, solved as a cubic.
For each real eigenvalue λ an eigenvector v solves (A − λI)v = 0. It is scaled so its largest entry is 1; any non-zero multiple is also an eigenvector.
Quick checks: the eigenvalues add up to the trace and multiply to the determinant. Real symmetric matrices always have real eigenvalues; rotations give complex ones.
Questions
What are the eigenvalues of [4, 1; 2, 3]?
λ² − 7λ + 10 = 0, so λ = 5 and λ = 2, with eigenvectors (1, 1) and (−1/2, 1).
Why are some eigenvalues complex?
The characteristic equation has no real roots, e.g. a rotation matrix [0, −1; 1, 0] gives λ = ±i.
Is the eigenvector unique?
No. Any non-zero multiple of it works. The calculator scales it so the largest entry is 1.
Why only 2×2 and 3×3?
These cover school and most engineering exercises and can be shown with clear working. Larger matrices need iterative numerical methods.
Is my data stored or sent anywhere?
No. The calculation runs entirely in your browser; nothing you type leaves your device.