Let \(A\) be an \(n \times n\) matrix and \(\lambda\) an eigenvalue of \(A\) with algebraic multiplicity \(m\text{.}\) The generalized eigenspace of \(A\) corresponding to \(\lambda\) is:
Understanding Generalized Eigenspaces: While regular eigenspaces consist of vectors \(\mathbf{v}\) such that \(A\mathbf{v} = \lambda\mathbf{v}\text{,}\) generalized eigenspaces include vectors that eventually become eigenvectors after applying \((A - \lambda I)\) enough times. These vectors form "chains" that give rise to the Jordan block structure.
