Eigenvalue and Eigenvector Calculator
This eigenvalue calculator finds the eigenvalues and eigenvectors of a square matrix from 2×2 up to 5×5. Type the matrix row by row, separating rows with semicolons or new lines, for example 4 1; 2 3. It builds the characteristic polynomial det(A − λI), solves it for every eigenvalue including repeated and complex ones, and then finds a basis of eigenvectors for each real eigenvalue, scaled to small whole numbers when possible. It also shows the trace and determinant so you can check that they equal the sum and the product of the eigenvalues.
Eigenvalue Calculator
Find the eigenvalues and eigenvectors of a 2×2 to 5×5 matrix, with the characteristic polynomial.
How to use the Eigenvalue Calculator
- Type the matrix, one row per line or separated by ;.
- Use spaces or commas between entries.
- Read the eigenvalues and their multiplicity.
- Read the eigenvectors in the table.
Eigenvalue equation
Av = λv ⇔ (A − λI)v = 0 ⇒ det(A − λI) = 0. Sum of eigenvalues = trace(A); product of eigenvalues = det(A).
Features
- 2×2 to 5×5 matrices
- Characteristic polynomial
- Repeated and complex eigenvalues
- Eigenvectors in whole numbers
- Trace and determinant check
Frequently asked questions
What is an eigenvalue?
An eigenvalue λ of a matrix A is a number where Av = λv for some nonzero vector v. The matrix only stretches or flips v by the factor λ.
How do you find eigenvalues of a 2×2 matrix?
Solve λ² − (trace)λ + det = 0. For [[4, 1], [2, 3]], trace 7 and det 10 give λ² − 7λ + 10 = 0, so λ = 5 and λ = 2.
How do you find an eigenvector?
Solve (A − λI)v = 0 for each eigenvalue. For λ = 5 in [[4, 1], [2, 3]], −v₁ + v₂ = 0, so v = (1, 1).
Can eigenvalues be complex?
Yes. A rotation matrix such as [[0, −1], [1, 0]] has eigenvalues i and −i, because no real vector keeps its direction under a 90° turn.
What does a repeated eigenvalue mean?
The eigenvalue is a double root of the characteristic polynomial. It may have two independent eigenvectors, like the identity matrix, or just one, like [[2, 1], [0, 2]].