Eigenvalue and Eigenvector Calculator
Eigenvalue and Eigenvector Calculator works out Eigenvalues and Eigenvectors from Matrix, Eigenvector scaling and Decimals shown. The default figures on this page give eigenvalues of 5, 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 Eigenvalue and Eigenvector Calculator calculates
Finds the eigenvalues, eigenvectors, characteristic polynomial, multiplicities and diagonalisation of a square matrix, including complex eigenvalues.
How to use this Eigenvalue and Eigenvector Calculator
Enter Matrix, Eigenvector scaling and Decimals shown. Eigenvalues updates as you type, and “Use a worked example” loads checked figures you can adjust. 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).
2x2 with integer eigenvalues example
With Matrix: 4 1; 2 3, the calculator returns Eigenvalues: 5, 2, Characteristic polynomial: λ^2 - 7λ + 10, Largest real eigenvalue: 5. This example is checked against an independent source in the test suite.
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
How do I calculate eigenvalues and eigenvectors?
Finds the eigenvalues, eigenvectors, characteristic polynomial, multiplicities and diagonalisation of a square matrix, including complex eigenvalues. Open “How this is worked out” beside the result to see each step for your own figures.
What is the difference between an eigenvalue and an eigenvector?
On the figures this page opens with, the eigenvalues is 5, 2. No single number fits everyone: it follows Matrix, Eigenvector scaling and Decimals shown, so enter your own above and the answer updates as you type.
What is an eigenvalue for dummies?
Finds the eigenvalues, eigenvectors, characteristic polynomial, multiplicities and diagonalisation of a square matrix, including complex eigenvalues. On this page it is computed from Matrix, Eigenvector scaling and Decimals shown and shown as Eigenvalues, Eigenvectors and Characteristic polynomial.
How to calculate eigen values in a calculator?
It needs Matrix, Eigenvector scaling and Decimals shown. Nothing else is required, and the figures stay in your browser.
What information do I need for the Eigenvalue and Eigenvector Calculator?
You need Matrix, Eigenvector scaling and Decimals shown. The page opens with default figures that give eigenvalues of 5, 2, so you can see the scale before entering your own.
What does this Eigenvalue and 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.