Eigenspace

Definition: Eigenspace

Let $A\in F^{n\times n}$ be a square matrix.

The eigenspace of an eigenvalue $\lambda$ is the set of all eigenvectors which belong to $\lambda$ together with the zero vector.

$\{\vec{v}\in F^n\mid A\vec{v}=\lambda \vec{v}\}$

Theorem: Structure of Eigenspaces

The eigenspace of an eigenvalue $\lambda$ is a subspace of the vector space $F^n$.

PROOF

TODO

Definition: Geometric Multiplicity

The geometric multiplicity of an eigenvalue $\lambda$ is the dimension of its eigenspace.

NOTATION

$\mathrm{geo}(\lambda )$