eigensubspaces

An eigensubspace, or eigenspace, of a linear operator T on a finite‑dimensional vector space V over a field F, associated with an eigenvalue λ in F, is the set Eλ = {v ∈ V : T(v) = λv}. This set includes the zero vector and forms a subspace of V. Equivalently, Eλ = ker(T − λI).

The dimension of Eλ is called the geometric multiplicity of λ. It is always at least 1 for an eigenvalue, and it is at most the algebraic multiplicity of λ, which is the multiplicity of λ as a root of the characteristic polynomial det(T − λI).

Distinct eigenvalues yield linearly independent eigenvectors, and the eigenspaces for different eigenvalues intersect only in {0}. The eigenspace decomposition of V provides a structural view of T: V can be written as the direct sum of its eigenspaces if and only if T is diagonalizable; in that case the sum of the dimensions of all Eλ equals dim V.

Computationally, to obtain Eλ for a given λ, one solves (A − λI)x = 0, where A is a matrix representing T in a chosen basis. This kernel gives the eigenvectors corresponding to λ.

Notes:

- Eλ is always invariant under T.

- Over the real field, complex eigenvalues may occur; their eigenspaces are defined over the complex extension, and real invariant subspaces may arise from complex conjugate pairs.

- Eigenspaces provide a canonical means to understand the action of T and to simplify problems by reducing T to a simpler, diagonal form when diagonalizable.