Eigenvector Calculator
Eigenvector calculator works out Eigenvalues and Eigenvectors from Matrix, Eigenvector scaling and Decimals shown. The default figures on this page give eigenvectors of 5: (1, 1) | 2: (1, -2); change any entry and the result, the working and the sources update together.
Your figures
Updates as you typeHow this is worked out
10 steps
- Matrix is 2x2.
- [ 4 1 ]
- [ 2 3 ]
- Characteristic polynomial det(λI - A) = λ^2 - 7λ + 10 = 0.
- Roots of the characteristic polynomial give the eigenvalues.
- Each eigenvector spans the null space of A - λI, found by Gauss-Jordan elimination in the complex plane.
- λ = 5 (algebraic multiplicity 1, geometric multiplicity 1): (1, 1)
- λ = 2 (algebraic multiplicity 1, geometric multiplicity 1): (1, -2)
- A = P D P⁻¹ with D = 5 0; 0 2 and P = 1 1; 1 -2.
- Check: the eigenvalues sum to the trace (7) and multiply to the determinant (10).
Eigenvalues (raw)table
| real | imaginary | modulus | text |
|---|---|---|---|
| 5 | 0 | 5 | 5 |
| 2 | 0 | 2 | 2 |
Its 5 worked examples are re-run on every release, last on 15 September 2026. What changed
Sources and references
3 sources
What the eigenvector calculator calculates
Finds the eigenvalues, eigenvectors, characteristic polynomial, multiplicities and diagonalisation of a square matrix, including complex eigenvalues.
How to use this eigenvector calculator
Enter Matrix, Eigenvector scaling and Decimals shown. Eigenvalues updates as you type. Change one figure at a time to see what drives the result.
How the result is worked out
Open “How this is worked out” beside the result for the step-by-step calculation behind Eigenvalues, Eigenvectors and Characteristic polynomial.
The method follows Gilbert Strang, Introduction to Linear Algebra, ch. 6 (eigenvalues and eigenvectors), Golub & Van Loan, Matrix Computations, sec. 8.5 (the Jacobi eigenvalue method) and Wikipedia, Faddeev-LeVerrier algorithm (characteristic polynomial).
What you can compare
Change one input at a time to see how it affects Eigenvalues, Eigenvectors and Characteristic polynomial. The inputs you can vary are Matrix, Eigenvector scaling and Decimals shown.
Results are general information, not financial, tax, legal, medical or engineering advice. Check any decision that matters with a qualified professional.
See something wrong? Report a problem with this calculator
Frequently asked questions
What information do I need for the eigenvector calculator?
You need Matrix, Eigenvector scaling and Decimals shown. The page opens with default figures that give eigenvectors of 5: (1, 1) | 2: (1, -2), so you can see the scale before entering your own.
What does this eigenvector calculator show?
It shows Eigenvalues, Eigenvectors, Characteristic polynomial, Eigenvalues (raw), Order (n), Trace (sum of eigenvalues), Determinant (product of eigenvalues) and Spectral radius (largest |lambda|), with the working and any warnings for your entries.
Can I check how the eigenvector calculator result was calculated?
Yes. The “How this is worked out” section shows the calculation steps, and the sources section identifies the references used for the method.