λ Eigenvalue & Eigenvector Calculator
Eigenvalue & Eigenvector Calculator
Find all eigenvalues and eigenvectors of any square matrix. Shows the characteristic polynomial det(A−λI)=0 and solves (A−λI)v=0 for each eigenspace.
Eigenvalue & Eigenvector Calculator
Characteristic polynomial + eigenvector computation
✓ Step-by-Step✓ Up to 6×6✓ Free⚡ Loading SymPy Engine…
ℹ Supports square matrices from 2×2 to 6×6. Powered by math.js for high-precision numeric computation.
Matrix A
Quick Examples
Understanding Eigenvalues and Eigenvectors
For a square matrix A, a scalar λ is an eigenvalue and a non-zero vector v is its corresponding eigenvector if:
$$A\mathbf{v} = \lambda\mathbf{v}$$
This means when A acts on v, the result is just a scaling of v by λ. The vector v keeps its direction (or is reversed if λ < 0).
How to Find Eigenvalues
- Form the characteristic equation: det(A − λI) = 0
- Expand the determinant to get the characteristic polynomial
- Solve the polynomial for λ (the eigenvalues)
How to Find Eigenvectors
For each eigenvalue λ, solve the homogeneous system (A − λI)v = 0. The non-trivial solutions form the eigenspace.
Key Facts
- An n×n matrix has exactly n eigenvalues (counting multiplicity, possibly complex)
- tr(A) = sum of eigenvalues; det(A) = product of eigenvalues
- Symmetric matrices always have real eigenvalues
- The eigenvalues of a diagonal matrix are its diagonal entries
Frequently Asked Questions
Real matrices can have complex eigenvalues when the characteristic polynomial has no real roots. Complex eigenvalues always appear in conjugate pairs (a+bi and a−bi). They occur in rotation-like transformations.
A repeated (degenerate) eigenvalue has algebraic multiplicity greater than 1. The geometric multiplicity (dimension of eigenspace) may be less than algebraic multiplicity, making the matrix defective.
The characteristic polynomial is p(λ) = det(A − λI). Its roots are the eigenvalues. For an n×n matrix it is a degree-n polynomial.