Originalvektoren

Originalvektoren are vectors that reveal invariant directions of a linear map. Given a square matrix A, a nonzero vector v is an originalvektor if Av = λv for some scalar λ. The scalar λ is the eigenvalue associated with v. If λ = 0, v is in the kernel and Av = 0; otherwise v is stretched or contracted by λ. The set of all vectors v that satisfy (A − λI)v = 0 for a fixed λ forms the eigenspace E_λ, a linear subspace of the domain.

To compute them, one first finds the eigenvalues by solving det(A − λI) = 0. For each eigenvalue λ, solve (A − λI)v = 0 to obtain the eigenvectors (a basis for E_λ). If A has n linearly independent eigenvectors, the matrix A is diagonalizable: there exists an invertible P such that P⁻¹AP = D, where D is diagonal with the eigenvalues on its diagonal.

Important properties include that eigenvectors corresponding to distinct eigenvalues are linearly independent; for real symmetric matrices, eigenvectors for different eigenvalues are orthogonal. Geometric multiplicity (dimension of E_λ) is at least 1 and at most the algebraic multiplicity of λ. Applications span solving linear differential equations, stability analysis, principal component analysis, vibration analysis, and computer graphics.