Eigenspaces of a Matrix

For a matrix $ M $ having for eigenvalues $ \lambda_i $, an eigenspace $ E $ associated with an eigenvalue $ \lambda_i $ is the set of eigenvectors $ \vec{v_i} $ which have the same eigenvalue and the zero vector. That is to say the kernel (or nullspace) of $ M - I \lambda_i $.

For an eigenvalue $ \lambda_i $, calculate the matrix $ M - I \lambda_i $ (with I the identity matrix) (also works by calculating $ I \lambda_i - M $) and calculate for which set of vector $ \vec{v} $, the product of my matrix by the vector is equal to the null vector $ \vec {0} $

Example: The 2x2 matrix $ M = \begin {bmatrix} -1 & 2 \\ 2 & -1 \end {bmatrix} $ has eigenvalues $ \lambda_1 = -3 $ and $ \lambda_2 = 1 $, the computation of the proper set associated with $ \lambda_1 $ is $ \begin{bmatrix} -1 + 3 & 2 \\ 2 & -1 + 3 \end {bmatrix}. \begin {bmatrix} v_1 \\ v_2 \end {bmatrix} = \begin {bmatrix} 0 \\ 0 \end {bmatrix} $ which has for solution $ v_1 = -v_2 $. The eigenspace $ E_{\lambda_1} $ is therefore the set of vectors $ \begin{bmatrix} v_1 \\ v_2 \end{bmatrix} $ of the form $ a \begin{bmatrix} -1 \ \\ 1 \end{bmatrix}, a \in \mathbb {R} $. The vector space is written $ \text{Vect} \left\{ \begin{pmatrix} -1 \\ 1 \end{pmatrix} \right\} $