eigenvectorveld

An eigenvectorveld, or eigenvector field, is a vector field that assigns to each point x in a domain a nonzero eigenvector of a matrix-valued function A(x). In other words, for each x, there exists an eigenvalue λ(x) with A(x) v(x) = λ(x) v(x), where v(x) is the eigenvector associated with that eigenvalue. The domain is typically a subset of Euclidean space or a smooth manifold, and A is assumed to vary with x, often smoothly.

Existence and smoothness of an eigenvectorveld depend on the spectral properties of A. If A(x) depends smoothly on x and its eigenvalues are simple (no crossings) in a region, then it is possible to choose eigenvectors v(x) that depend smoothly on x locally. However, because eigenvectors are defined only up to a nonzero scalar, one must fix a normalization (and sometimes a sign or phase) to obtain a bona fide vector field. In cases where A(x) is real symmetric (or Hermitian in the complex case), eigenvectors corresponding to distinct eigenvalues are orthogonal, which simplifies both the construction and interpretation of the field.

Global existence can be hindered by topology. Even when local smooth eigenvectorfields exist, a global, nonvanishing eigenvectorveld may be obstructed by topological invariants, such as those related to line bundles and eigenvalue degeneracies.

Applications and examples: in physics and engineering, A(x) often represents a tensor field (such as a diffusion, stiffness, or inertia tensor), and the eigenvectorveld identifies principal directions of anisotropy or deformation. In diffusion tensor imaging, principal eigenvector fields indicate dominant diffusion directions. Computationally, eigenvector fields are computed pointwise, with additional care to maintain continuity across x near eigenvalue crossings.

See also: eigenvalues, eigenvectors, tensor fields, spectral theory, principal directions.