Eigenvalues & Eigenvectors

About this calculator

The Eigenvalues & Eigenvectors Calculator finds the eigenvalues and eigenvectors of any real 2×2 or 3×3 matrix by solving the characteristic polynomial det(A − λI) = 0 exactly in high-precision decimal arithmetic. It reports each eigenvalue with its algebraic and geometric multiplicity, a basis for its eigenspace, the trace, determinant, and characteristic polynomial. Complex conjugate pairs, repeated roots, and defective matrices (too few independent eigenvectors) are all detected and labeled.

How to use the Eigenvalues & Eigenvectors calculator

1. Enter the values for your problem into the input fields.
2. Read the result — it updates instantly as you type.
3. Check the formula and the visual explanation to follow how the answer was found.
4. Copy the page URL to share the exact calculation.

Common examples

• Symmetric [[2,1],[1,2]] → λ = 3 with eigenvector (1, 1) and λ = 1 with (1, −1)
• Rotation [[0,−1],[1,0]] → complex pair λ = ±i, no real eigenvectors
• Shear [[3,1],[0,3]] → repeated λ = 3, defective: geometric multiplicity 1 < algebraic 2
• Diagonal 3×3 diag(2, 3, 4) → λ = 4, 3, 2 with the standard basis vectors as eigenvectors
• [[4,1],[2,3]] → trace 7, det 10, eigenvalues λ = 5 and λ = 2