Eigenspacen

Eigenspacen is a term that appears as a variant or nonstandard form of the more common mathematical concept eigenspace. In standard usage, an eigenspace is a subspace associated with a linear transformation that contains all vectors that are scaled by a fixed scalar (an eigenvalue) under that transformation.

For a linear transformation T on a finite-dimensional vector space V, an eigenvector v ≠ 0 satisfies T(v) = λv for some scalar λ, known as an eigenvalue. The eigenspace corresponding to λ is defined as Eλ = { v ∈ V | T(v) = λv }. Equivalently, Eλ is the null space of T − λI, and thus a subspace of V that includes all eigenvectors for λ and the zero vector.

Key properties include: the dimension of Eλ, called the geometric multiplicity, satisfies 1 ≤ dim(Eλ) ≤ the algebraic multiplicity of λ; eigenvectors corresponding to distinct eigenvalues are linearly independent; a linear transformation is diagonalizable if the direct sum of its eigenspaces equals V. The eigenspaces provide a spectral decomposition when a basis of eigenvectors exists.

Computationally, eigenvalues are found by solving det(A − λI) = 0 for a matrix A representing T; for each eigenvalue λ, solve (A − λI)x = 0 to obtain a basis for Eλ.

Applications span diagonalization, spectral decomposition, principal component analysis, solving differential equations, and stability analysis. Etymologically, "eigen" means characteristic in German, and "eigenspace" is the standard term; "eigenspacen" may reflect a nonstandard plural form in some contexts. See also eigenvalue, eigenvector, and diagonalization.