Eigendirection

Eigendirection is a term used in linear algebra and related fields to describe the direction in which a linear transformation acts as a simple scaling operation. It commonly refers to eigenvectors, which are non‑zero vectors that, when multiplied by a linear transformation matrix, result in a scalar multiple of themselves. The scalar factor is called the eigenvalue. If a linear operator \(T\) acting on a vector space \(V\) satisfies \(T(v)=\lambda v\), then \(v\) is an eigenvector of \(T\) with eigenvalue \(\lambda\), and \(v\)’s line through the origin gives the eigendirection associated with \(\lambda\).

In more geometric terms, an eigendirection represents the orientation preserved by the transformation, while the eigenvalue indicates how the transformation stretches or contracts along that direction. If \(\lambda\) is positive, the direction is preserved; if \(\lambda\) is negative, the direction is reversed. An eigenvalue of zero implies that the associated eigendirection collapses to the zero vector, representing a degenerate or scaling direction.

Eigendirections are fundamental to many applications, including diagonalization of matrices, stability analysis in differential equations, and principal component analysis in statistics. They also play a role in differential geometry through the study of eigenvalues of the Laplace–Beltrami operator, where eigendirections correspond to orthogonal modes of vibration or heat diffusion on a manifold.