eigenspacet

Eigenspacet is not a standard term in mathematics. It is sometimes encountered as a variant spelling or neologism intended to refer to the subspace associated with a fixed eigenvalue of a linear operator. In most mathematical literature, the object is called an eigenspace, and the article below presents a conventional interpretation while noting the term’s nonstandard status.

Let T be a linear operator on a vector space V over a field F. For a scalar λ in F, the eigenspacet associated with λ is the set E_λ(T) = { v ∈ V : T(v) = λ v }. This set is a subspace of V and is equal to the kernel of (T − λI). The dimension of E_λ(T) is called the geometric multiplicity of the eigenvalue λ; the eigenvalue’s algebraic multiplicity is obtained from the characteristic polynomial det(T − λI). Distinct eigenvalues have distinct, complementary eigenspaces when V decomposes accordingly.

In finite dimensions, if V is isomorphic to F^n and T is represented by a matrix A, E_λ(T) can be found by solving (A − λI)x = 0. The solutions form a basis for the eigenspacet. The sum of the dimensions of E_λ(T) over all distinct eigenvalues λ equals the dimension of V.

Example: For A = [[2, 0], [0, 3]] acting on F^2, the eigenspacet for λ = 2 is span{(1, 0)} and for λ = 3 is span{(0, 1)}.

Generalizations and related concepts include eigenvectors in tensor spaces and in infinite-dimensional settings under spectral theory. When used outside standard context, eigenspacet is typically synonymous with the eigenspace. See also eigenspace, eigenvalue, eigenvector, Jordan form, and the spectral theorem.