Eigenvalue and eigenvector calculator
Enter your values. The answer and the full worked solution update as you type.
Result
| Eigenvalue | Multiplicity | Eigenvector | Unit eigenvector |
|---|---|---|---|
| 5 | 1 | (2, 0, 1) | (0.8944, 0, 0.4472) |
| 3 | 1 | (1, 1, −1) | (0.5774, 0.5774, −0.5774) |
| 2 | 1 | (1, 0, −1) | (0.7071, 0, −0.7071) |
Find the eigenvalues and eigenvectors of a 2×2 or 3×3 matrix. The calculator expands det(A − λI) into the characteristic polynomial, solves it for real or complex eigenvalues with their multiplicity, and finds the eigenvectors of every real eigenvalue by solving (A − λI)v = 0. Each eigenvector is checked by computing A·v, and for repeated eigenvalues it shows whether the matrix can be diagonalised.
Step-by-step solution
Characteristic equation det(A − λI) = 0
Eigenvalues
Eigenvector (λ = 5)
Eigenvector (λ = 3)
Eigenvector (λ = 2)
Key formulas
- Eigenvalues
- det(A − λI) = 0
How to find eigenvalues and eigenvectors
- Subtract λ from every diagonal entry of A to get A − λI.
- Set the determinant to zero: det(A − λI) = 0. For a 2×2 matrix this is λ² − (tr A)λ + det A = 0; for a 3×3 matrix it is a cubic equation.
- Solve the characteristic equation. Check: the eigenvalues add up to the trace and multiply to the determinant.
- For each eigenvalue, solve (A − λI)v = 0 by row reduction. The solutions form a line (or a plane); any non-zero vector in it is an eigenvector.
- Check every eigenvector: A·v must equal λ·v.
Questions and answers
How do I enter the matrix?
Type the rows separated by semicolons and the numbers in each row separated by spaces, for example 4 1 2; 0 3 0; 1 -1 3 for a 3×3 matrix. Decimals and negative numbers are allowed.
Why can a matrix have complex eigenvalues?
The characteristic polynomial can have complex roots, and for a real matrix they always come in conjugate pairs. The rotation matrix 0 -1; 1 0 has the eigenvalues ±i, because no real vector keeps its direction under a rotation.
What does a repeated eigenvalue mean?
A root of the characteristic polynomial with multiplicity 2 or 3. The matrix may still have that many independent eigenvectors, like the identity matrix, or fewer. With fewer it is defective and cannot be diagonalised.
Is the eigenvector unique?
No. Any non-zero multiple of an eigenvector is also an eigenvector. The calculator shows a simple form, with whole numbers where possible, and the unit vector of length 1.
What are eigenvalues used for?
They give the natural frequencies of vibrating systems, the principal stresses of a stress tensor, the stability of systems of differential equations and the main directions in principal component analysis.
Practise it yourself
Practise Matrices & determinants
A calculator gives the answer. In the exam you need the method: practise it with new problems, hints and worked solutions.