Orthonormalny

Orthonormalny, or orthonormal, is a term used in linear algebra to describe a set of vectors that are mutually orthogonal and of unit length with respect to an inner product. Specifically, a set {e1, e2, ..., en} is orthonormal if the inner product satisfies ⟨ei, ej⟩ = δij, where δij is the Kronecker delta. This means ⟨ei, ei⟩ = 1 and ⟨ei, ej⟩ = 0 for i ≠ j.

An orthonormal basis is a basis consisting of orthonormal vectors. In an n-dimensional space with the standard inner product, the coordinates of a vector x relative to an orthonormal basis are xi = ⟨x, ei⟩, and x can be written as x = sum_i ⟨x, ei⟩ ei. The norm of x is ||x|| = sqrt(⟨x, x⟩), which equals the Euclidean norm in the common real case.

Matrix interpretation is convenient: the columns of a real matrix Q are orthonormal if Q^T Q = I, in which case Q is called an orthogonal matrix. In complex spaces, columns are orthonormal when Q^* Q = I, making Q unitary. The Gram–Schmidt process converts any linearly independent set into an orthonormal set, producing an orthonormal basis for the span.

Projections are simplified with an orthonormal set: the projection of v onto the span of {u1, ..., uk} is sum_i ⟨v, ui⟩ ui. Orthornormal bases underpin many algorithms in numerical linear algebra, such as QR decomposition, and appear in applications including signal processing, computer graphics, and Fourier analysis. In Hilbert spaces, a complete orthonormal set provides a basis for representing vectors, with identities like Parseval generalizing the idea.