eigenspaces

An eigenspace of a linear operator T on a vector space V over a field F is the set of all vectors v in V such that T(v) = λ v for a given eigenvalue λ. Equivalently, E_λ(T) is the kernel of T − λI. It is a subspace of V.

For a matrix A, eigenvalues are found from the characteristic equation det(A − λI) = 0. For each eigenvalue λ, the associated eigenspace E_λ(A) is ker(A − λI). Its dimension is the geometric multiplicity of λ, and satisfies 1 ≤ dim E_λ ≤ the algebraic multiplicity m_λ. The sum of the dimensions of all E_λ is at most the dimension of V, and equals that dimension when the matrix is diagonalizable.

Eigenspaces corresponding to distinct eigenvalues intersect only in the zero vector. If the operator is diagonalizable, V is the direct sum of its eigenspaces, and the operator is similar to a diagonal matrix. For real symmetric matrices, the eigenspaces are orthogonal.

Example: Consider A = [[2, 1], [0, 3]]. The eigenvalues are λ = 2 and λ = 3. For λ = 2, A − 2I = [[0, 1], [0, 1]]; solving (A − 2I)v = 0 gives y = 0, so E_2 = span{(1, 0)}. For λ = 3, A − 3I = [[-1, 1], [0, 0]]; solving gives y = x, so E_3 = span{(1, 1)}. The two eigenspaces are independent, so A is diagonalizable.