Compute the eigenvalues and eigenvectors of a square matrix $A$. This calculator finds both standard and generalized eigenvectors when needed.
Supports real, complex, fractions (e.g. 3+2i, 1/2), and expressions like pi, sqrt(2), exp(1).
Eigenvalues and Eigenvectors:
For a square matrix $A$, an eigenvalue $\lambda$ and corresponding eigenvector $\mathbf{v}$ satisfy:
$$ A\mathbf{v} = \lambda\mathbf{v} $$
The eigenvector $\mathbf{v}$ is a direction that gets scaled by factor $\lambda$ when transformed by $A$.
Generalized Eigenvectors:
When a matrix is defective (lacks enough eigenvectors to span the space), we need generalized eigenvectors $\mathbf{w}$ that satisfy:
$$ (A - \lambda I)^k\mathbf{w} = \mathbf{0} \quad \text{for some } k > 1 $$
Generalized eigenvectors complete the basis when geometric multiplicity is less than algebraic multiplicity.