Find Eigenvectors Calculator
Calculate the real or complex eigenvalues of a real 2×2 matrix, plus representative unit eigenvectors when real directions exist.
Written and checked by TheToolNet Editorial Team · Last reviewed: September 20, 2026 · Editorial method
Find eigenvalues and eigenvectors
How eigenvalues are calculated for 2×2 matrices
For A = [[a,b],[c,d]], solve det(A−λI)=0:
The trace a+d is the sum of eigenvalues and determinant ad−bc is their product. For each real λ, solve (A−λI)v=0 for a nonzero eigenvector.
Repeated and complex cases
A repeated eigenvalue may have one eigendirection or an entire plane of eigenvectors. A real 2×2 matrix with negative discriminant has a complex-conjugate eigenvalue pair and no real eigenvector directions. Explore concrete cases in the worked examples, or use the matrix diagonalization calculator to check whether enough independent eigenvectors exist.