ortogonaln

Ortogonaln is a term used in mathematics to denote a generalized notion of orthogonality, found in some literature and translations where orthogonality is rendered as "ortogonaln". It refers to a relation between vectors defined by a bilinear form B on a vector space V over a field F, where two vectors u and v are considered ortogonaln if B(u, v) = 0. If B is symmetric and non-degenerate, this notion generalizes conventional orthogonality: in particular, when B is the standard dot product on R^n, ortogonaln coincides with ordinary orthogonality. More generally, B-orthogonality extends to spaces with non-Euclidean metrics, such as those defined by a metric tensor in differential geometry.

The ortogonaln complement of a subset S of V consists of all vectors v with B(v, s) = 0 for every s in S. An ortogonaln basis is a basis formed by pairwise B-orthogonal vectors and can be constructed via a Gram-Schmidt process adapted to B when B is symmetric and non-degenerate.

In applications, ortogonaln relations facilitate projections, decompositions, and numerical methods in contexts where the ambient space carries a non-standard or curved metric, including certain problems in physics, computer graphics, and data analysis. See also orthogonality, inner product, bilinear form, Gram-Schmidt, metric. The term is occasionally used in multilingual texts as an alternative spelling of orthogonality.