Eigenvector Calculator
Solve (A − λI)v = 0 step by step for each real eigenvalue and get a clean basis for its eigenspace - live as you type.
Exact eigenvectors are shown for real eigenvalues. Complex eigenvalues are not currently supported.
Solution
Eigenvalues of A
Eigenvectors
Check
Runs entirely on your device - nothing is ever uploaded.
More tools
- Eigenvalue Calculator Find eigenvalues and see the characteristic polynomial for a matrix. Get started
- Dot Product Calculator Dot two vectors with every product summed, plus magnitudes and the angle. Get started
- Inverse Matrix Calculator Invert a square matrix and verify it multiplies back to I. Get started
- Derivative Calculator Differentiate with steps, up to the second derivative. Get started
- Determinant Calculator Determinant of a square matrix, from 1×1 through 6×6. Get started
- Integral Calculator Antiderivatives with the technique behind every step. Get started
- Matrix Multiplication Calculator Multiply two matrices with step-by-step working for every cell. Get started
- Quadratic Formula Calculator Roots in exact radicals, discriminant, and vertex. Get started
How to use the eigenvector calculator
- Pick 2×2 or 3×3 and type a number (decimals and fractions like 1/5 both work) into every cell.
- The eigenvalues are found from det(A − λI) = 0 automatically.
- For each real eigenvalue, rational or square-root, the calculator solves (A − λI)v = 0 by exact elimination and shows a basis.
- The check line multiplies A by the vector so you can verify A v = λ v yourself.
Eigenvectors are not unique: any nonzero multiple works. Each basis vector is normalized to a 1 in its free variable.
Why eigenvectors matter
An eigenvector is a direction a matrix only stretches, never rotates - the eigenvalue gives the stretch factor. That property powers facial recognition, vibration analysis, and quantum states, where each measurable quantity is an eigenvalue of its operator. Because any nonzero multiple of an eigenvector is still an eigenvector, this calculator normalizes each basis vector to a leading 1 so equivalent answers always look the same.