eigenspace

The eigenspace is the span of all eigenvectors associated with a particular eigenvalue, along with the zero vector.

The eigenspace $S_i$ contains the eigenvectors associated with ${\lambda }_i$:

$$S_i=\{\mathbf{x}\in {\mathbb{R}}^n|A\mathbf{x}={\lambda }_i\mathbf{x}\}$$

If ${\lambda }_i\ne {\lambda }_j$ then $S_i$ and $S_j$ are mutually orthogonal, i.e. every vector of $S_i$ is orthogonal to every vevctor of $S_j$.

If all eigenvectors are orthogonal to eachother (e.g. the case for real symmetric matrices) they can be normalized to form an orthonormal basis.