Orthogonal

Orthogonal describes a relationship of perpendicularity in Euclidean space, but also a generalized notion of independence in inner product spaces. In geometry, two lines or vectors are orthogonal if they meet at a right angle, which corresponds to their dot product being zero in standard coordinates. In an abstract inner product space, two elements x and y are orthogonal when their inner product ⟨x, y⟩ equals zero.

In linear algebra, a set of vectors is orthogonal if every pair is orthogonal; if each vector has unit length, the set is orthonormal. An orthogonal (or orthonormal) basis simplifies projections and decompositions. A square matrix is orthogonal if its transpose equals its inverse (A^T A = I); then its columns (and rows) form an orthonormal set and the matrix preserves lengths and angles.

Orthogonality is central in functional spaces: functions f and g are orthogonal on a domain D if their inner product ∫_D f(x) g(x) dx equals zero. This underlies Fourier series, Legendre polynomials, and other orthogonal systems. The Gram–Schmidt process converts any basis into an orthogonal (or orthonormal) one.

In statistics and data analysis, orthogonality expresses independence of effects or contrasts in experimental design; orthogonal designs help ensure unconfounded estimates. In signal processing and data analysis, orthogonalization reduces redundancy among features, as in principal component analysis producing orthogonal components.

Etymology and nuance: the term reflects the idea of straight angles or unconfounded directions; note that “orthogonal” is most precise in spaces equipped with a chosen inner product, and its practical meaning aligns with perpendicularity in that context.