Principalretninger

Principalretninger, or principal directions, are directions in which a certain quantity is extremal or decoupled under a linear operator or geometric context. In geometry and data analysis, they identify natural axes along which the behavior is simplest.

In differential geometry, for a smooth surface in 3D, the principal directions at a point are the eigenvectors of the shape operator. They indicate the directions along which normal curvature attains its maximum and minimum. The corresponding principal curvatures are the eigenvalues. For a regular surface, the two principal directions are orthogonal; if the surface is umbilic (as on a sphere), the principal curvatures are equal and the directions are not unique.

In linear algebra, the principal directions of a symmetric matrix A are its eigenvectors. They define the principal axes of the associated quadratic form and provide a coordinate system in which the form is diagonal. This diagonalization corresponds to a change of basis that decouples the variables.

In statistics and data analysis, principal directions arise in principal component analysis (PCA). The principal directions are the eigenvectors of the covariance matrix and indicate directions of maximal variance in the data. Projections onto these directions reduce dimensionality while preserving most of the variability.

Applications span computer graphics, structural mechanics, material science, and multivariate statistics. The concept emphasizes choosing coordinate directions that simplify the description of curvature, energy, or variability. The term principalretninger is Danish and Norwegian for “principal directions.”