eigenrummet

Eigenspace, known in Danish as eigenrummet, is the subspace of vectors that are scaled by a fixed factor under a linear transformation. For a square matrix A over a field F and a scalar λ in F, the eigenspace associated with λ is denoted Eλ and defined as Eλ = { v in F^n : Av = λv } = ker(A − λI). This set includes the zero vector and all eigenvectors corresponding to λ, forming a linear subspace of F^n.

The eigenrummet can be understood as the kernel of A − λI. If λ is not an eigenvalue of A, then Eλ = {0}. If λ is an eigenvalue, Eλ consists of all nonzero eigenvectors for which Av = λv, and its dimension is called the geometric multiplicity of λ. This dimension is at least 1 and at most the algebraic multiplicity of λ, and it can be smaller than the multiplicity when eigenvectors do not span all directions associated with λ.

Eigenspaces play a central role in diagonalization and spectral theory. A matrix is diagonalizable if the direct sum of its eigenspaces over all distinct eigenvalues equals the entire space, meaning there are enough linearly independent eigenvectors to form a basis. The eigenrummet for each eigenvalue provides the invariant subspace structure of the transformation, and together they reveal how the transformation acts by simple scaling along these subspaces.

Example: A = [[4, 1], [0, 4]] has eigenvalue λ = 4 with A − 4I = [[0, 1], [0, 0]]. The nullspace of this matrix is spanned by (1, 0)^T, so E4 = span{(1, 0)}.