Find the real eigenvalues and corresponding eigenvectors of a 2×2 or 3×3 matrix.
Understanding the eigenvector calculation
An eigenvector is a direction the matrix does not rotate — it only stretches it, by a factor called the eigenvalue. Finding them means solving the characteristic polynomial and then the null space of A − λI for each root.
The calculator solves the characteristic equation exactly for 2×2 matrices and numerically for 3×3, then row-reduces to extract a unit eigenvector for each real eigenvalue.
A worked example
Take the starting scenario, a square matrix — 2×2 or 3×3, one row per line of 4 1 2 3. That gives a eigenvalues of 5, 2. Underneath, Eigenvalue 1 comes out at 5 and Eigenvector 1 at (0.7071, 0.7071). Change any field and every figure updates as you type.
How to fill it in
- Enter a square matrix, one row per line.
- Read the characteristic polynomial and eigenvalues.
- Read the unit eigenvector for each eigenvalue.
Formula reference
characteristic equation: det(A − λI) = 0
eigenvector: any non-zero v with (A − λI)v = 0
trace = sum of eigenvalues · determinant = product of eigenvalues
Questions people ask about this
What is an eigenvector used for?
Principal component analysis, stability of differential equations, vibration modes and page ranking all reduce to eigenvector problems.
Why are some eigenvalues complex?
Rotation matrices genuinely have no real invariant direction. This calculator reports real eigenvalues and flags when the pair is complex.
Are eigenvectors unique?
Only up to scale — any multiple of an eigenvector is also one. The calculator normalises to unit length by convention.