Eigenvalues & Eigenvectors Calculator

Compute all eigenvalues and eigenvectors of any square matrix (2x2 to 5x5) with algebraic and geometric multiplicities, full characteristic polynomial, and RREF-based null space computation.

Matrix Setup

Size (n x n)

Theory & Definitions

Eigenvalue equation: $Av = \lambda v$ where $v \neq 0$. Equivalently: $(A - \lambda I)v = 0$.

Characteristic polynomial: $p(\lambda) = \det(A - \lambda I) = 0$. The roots are the eigenvalues.

Algebraic multiplicity (AM): The multiplicity of $\lambda$ as a root of $p(\lambda)$.
Geometric multiplicity (GM): The dimension of the null space of $(A - \lambda I)$ = number of linearly independent eigenvectors for $\lambda$.
Always: $1 \leq \text{GM}(\lambda) \leq \text{AM}(\lambda)$.

Diagonalisable: A matrix is diagonalisable if and only if GM = AM for every eigenvalue, i.e. the total number of linearly independent eigenvectors = n.