Finding eigenvectors

To find the eigenvectors for a particular eigenvalue λ, you need to solve the associated homogeneous linear system (A – λI)x = 0. Since the solutions to such systems form a subspace of Rn, what you're doing is finding a subspace of eigenvectors.

The eigenspace of a square matrix A corresponding to the eigenvalue λ is the subspace containing the zero vector and all the eigenvectors of A with eigenvalue λ.

To describe the eigenspace, you find a basis of it.

To find a basis of an eigenspace:

For a previously found eigenvalue λ of a matrix A:

find a basis of the solution space of the homogeneous linear system (A – λI)x = 0.