Eigenvector Calculator
Enter a 2×2 or 3×3 matrix and get its eigenvalues and eigenvectors with the full derivation — characteristic equation, roots, and the row reduction behind each eigenvector.
An eigenvector keeps its direction when A is applied — only its length changes, by the eigenvalue. λ = 1, λ = 3.
Set up the characteristic equation
Eigenvalues λ are the scalars for which A − λI loses its inverse — the matrix squashes space onto a lower dimension, so its determinant is zero.
2 1 1 2 A
det(A − λI) = 0 → λ² − 4λ + 3 = 0Solve for the eigenvalues
A 2×2 gives a quadratic — solve it with the formula.
- Real eigenvalue
- λ = 1
- Real eigenvalue
- λ = 3
Eigenvectors for λ = 1
Solve (A − λI)v = 0: row-reduce, then read the null space. Any nonzero multiple of an eigenvector is also an eigenvector.
1 1 1 1 A − λI, ready to row-reduce
- v (any t ≠ 0)
- t · [-1, 1]
Eigenvectors for λ = 3
Solve (A − λI)v = 0: row-reduce, then read the null space. Any nonzero multiple of an eigenvector is also an eigenvector.
-1 1 1 -1 A − λI, ready to row-reduce
- v (any t ≠ 0)
- t · [1, 1]
Works instantly in your browser — nothing is uploaded, and the URL keeps your numbers so you can share the exact worked example.
Studying statistics or linear algebra? See every tool in the Stats & Algebra hub — variance, interpolation, RREF and eigenvectors, cross-linked.The geometric picture
Most vectors change direction when a matrix is applied to them. Eigenvectors are the stubborn ones: they come back pointing along the same line, scaled by their eigenvalue. A positive λ stretches, a negative λ flips, and |λ| > 1 grows. Finding them means finding the matrix's natural axes.