n×northogonale

n×northogonale is a term used to describe matrices that are orthogonal with respect to a weighted inner product on R^n. Specifically, for a fixed positive diagonal matrix W, a real n×n matrix A is called northogonal (with respect to W) if A^T W A = I_n. The weight matrix W encodes a directional bias in the inner product, so the orthogonality condition depends on this metric. When W = I_n, the concept reduces to ordinary orthogonality and A lies in the standard orthogonal group O(n).

A common way to generate northogonal matrices for a given W is to take A = W^{-1/2} Q, where Q is any ordinary orthogonal matrix. Then A^T W A = Q^T Q = I_n. This shows that the set of northorthogonal matrices relative to W forms a group under multiplication, denoted O_W(n). The group O_W(n) is isomorphic to the ordinary orthogonal group O(n) via the transformation A ↦ W^{1/2} A W^{-1/2}.

Key properties follow from this structure. If A ∈ O_W(n) then det(A) = ±1 / √det(W), so the determinant’s magnitude is fixed by the weight W. The concept generalizes to different positive definite W, enabling a family of weighted orthogonality conditions for diverse metric choices.

In practice, northogonality arises in contexts such as weighted least squares, anisotropic numerical linear algebra, and multivariate statistics, where a directional metric better reflects the data or material properties than the Euclidean one. The term “northogonale” is not universally standardized and is often described in terms of A^T W A = I or as the weighted orthogonal group O_W(n).

See also: orthogonal matrix, weighted inner product, Gram–Schmidt, weighted least squares.