Eigenvector Calculator

For a square matrix A, an eigenvector is a non-zero vector v such that Av = λv for some scalar λ (the eigenvalue). Multiplying by A only stretches v by λ; it does not change v's direction. Eigenvectors describe the matrix's natural axes.

First find the eigenvalues by solving det(A − λI) = 0 (the characteristic equation). Then for each eigenvalue λ, solve the linear system (A − λI)v = 0 for v. The solution is a one-dimensional subspace; pick any non-zero vector in it. The calculator returns the unit-length form.

For A = [[a, b], [c, d]], the characteristic equation is λ² − (a+d)λ + (ad−bc) = 0. That is λ² − trace(A) · λ + det(A) = 0. The discriminant is trace² − 4·det, which decides whether eigenvalues are real and distinct, repeated, or a complex conjugate pair.

If Av = λv, then A(kv) = λ(kv) for any non-zero scalar k. So any non-zero multiple of an eigenvector is also an eigenvector. To pick one specific representative, conventions usually use the unit eigenvector (length 1) or set one coordinate to 1.

Then the matrix has a complex conjugate pair of eigenvalues. The corresponding eigenvectors are also complex-valued. This calculator reports the complex eigenvalues but does not produce complex eigenvectors. Use a dedicated linear-algebra package (NumPy, SymPy, MATLAB, etc.) for the complex case.

The matrix has a single repeated eigenvalue. There is at most a one-dimensional eigenspace for it; the calculator returns one eigenvector for the repeated eigenvalue. The matrix may also be defective (no second linearly independent eigenvector), but that distinction needs the Jordan form, not just the characteristic polynomial.

No. This calculator is scoped to 2×2 real matrices. For 3×3 and larger, the characteristic polynomial becomes high-degree and root-finding requires numerical methods (QR algorithm, power iteration, etc.). Use a CAS or scientific library for those.