Matrix Algebra

Eigenspace

Eigenspace. Given an eigenvalue $\lambda$ , the corresponding eigenvector $\mathbf{x}$ must satisfy the homogeneous equation

$\displaystyle (A - \lambda I)\mathbf{x} = \mathbf{0}$

Thus, the null space of $(A - \lambda I)$ represents the collection of eigenvalues corresponding to $\lambda$ , and $\mathrm{null}\,(A - \lambda I)$ is called the eigenspace. The dimension of the eigenspace $\mathrm{null}\,(A - \lambda I)$ is called the geometric multiplicity of the eigenvalue $\lambda$ .

EXAMPLE 2. Find a basis for the eigenspace corresponding to the value $\lambda$ in each of the following.

$A = \begin{bmatrix} 1 & 2 & 2 \\ 3 &-2 & 1 \\ 0 & 1 & 1 \end{bmatrix}$ with $\lambda = 3$ .
$A = \begin{bmatrix} 1 & 0 &-1 \\ 1 &-3 & 0 \\ 4 &-13& 1 \end{bmatrix}$ with $\lambda = -2$ .
$A = \begin{bmatrix} 3 & 0 & 2 & 0 \\ 1 & 3 & 1 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 4 \\ \end{bmatrix}$ with $\lambda = 4$ .