Eigenvalue and Eigenvector Calculator
Find eigenvalues, eigenvectors, the characteristic polynomial, diagonalization status, and step-by-step work for 2×2 and 3×3 matrices.
Background
Eigenvalues and eigenvectors describe the special directions of a matrix transformation. An eigenvector keeps its direction after the transformation, while its eigenvalue tells you how much that direction is stretched, shrunk, or flipped.
How to use this Eigenvalue and Eigenvector Calculator
- Choose a 2×2 or 3×3 matrix.
- Enter values directly into the grid, or paste a square matrix into the paste box.
- Click Calculate eigenvalues to see eigenvalues, eigenvectors, and the characteristic polynomial.
- Use the options to show or hide steps, eigenvector bases, and the diagonalization check.
- Use quick examples to explore diagonal, symmetric, repeated, defective, and complex-eigenvalue cases.
How this calculator works
- The calculator forms the characteristic equation det(A − λI) = 0.
- For 2×2 matrices, it solves the quadratic equation exactly when possible.
- For 3×3 matrices, it builds the cubic characteristic polynomial and solves it numerically.
- For each real eigenvalue, it solves (A − λI)v = 0 to find an eigenvector basis.
- It checks whether the matrix has enough independent eigenvectors to be diagonalizable.
Formula & Concepts Used
Eigenvalue equation: Av = λv, where v is a nonzero eigenvector.
Characteristic equation: det(A − λI) = 0.
Eigenspace: the solution space of (A − λI)v = 0 for a specific eigenvalue.
Algebraic multiplicity: how many times an eigenvalue appears as a root of the characteristic polynomial.
Geometric multiplicity: the number of independent eigenvectors for an eigenvalue.
Diagonalizable matrix: a matrix with enough independent eigenvectors to write A = PDP⁻¹.
Example Problems & Step-by-Step Solutions
Example 1: 2×2 matrix with two real eigenvalues
Find the eigenvalues and eigenvectors of:
[2 1][1 2]
The characteristic equation is λ² − 4λ + 3 = 0, so the eigenvalues are λ = 3 and λ = 1.
Example 2: Repeated eigenvalue
Find the eigenvalues of:
[2 1][0 2]
The only eigenvalue is λ = 2 with algebraic multiplicity 2. Because it has only one independent eigenvector, the matrix is defective.
Example 3: Complex eigenvalues
Find the eigenvalues of a rotation matrix:
[0 −1][1 0]
The characteristic equation is λ² + 1 = 0, so the eigenvalues are λ = i and λ = −i. There are no nonzero real eigenvectors.
FAQs
What is an eigenvalue?
An eigenvalue is the scale factor applied to an eigenvector when a matrix transforms that vector.
What is an eigenvector?
An eigenvector is a nonzero vector whose direction does not change under a matrix transformation. It may stretch, shrink, or flip direction.
How do you find eigenvalues?
Set up and solve the characteristic equation det(A − λI) = 0.
How do you find eigenvectors?
For each eigenvalue, solve the homogeneous system (A − λI)v = 0.
What does diagonalizable mean?
A matrix is diagonalizable when it has enough linearly independent eigenvectors to form a basis. In that case, it can be written as A = PDP⁻¹.
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