orthonormale

Orthonormal, often written as orthonormal, describes a set of vectors in an inner product space that are mutually orthogonal and of unit length. In a finite-dimensional space V with an inner product ⟨·,·⟩, a set {u1, ..., um} is orthonormal if ⟨ui, uj⟩ = δij for all i, j, where δij is the Kronecker delta. In Euclidean space with the standard inner product, this means ui · uj = 0 for i ≠ j and ||ui|| = 1.

An orthonormal basis is a maximal orthonormal set that spans the space. In R^n, any orthonormal basis consists of n vectors, and every vector x ∈ R^n can be expressed as x = sum_i ⟨x, ui⟩ ui. The squared norm satisfies ||x||^2 = sum_i ⟨x, ui⟩^2. The projection of a vector onto the span of an orthonormal set is given by the sum of its inner products with the basis vectors, i.e., proj = sum_i ⟨x, ui⟩ ui.

In matrix form, a matrix Q with orthonormal columns satisfies Q^T Q = I, and if Q is square, it is called orthogonal (over the real case) or unitary (over the complex case). Orthogonal matrices represent rotations and reflections and preserve inner products and norms. The Gram–Schmidt process constructs an orthonormal set from any linearly independent set, enabling QR decompositions and simplified geometric interpretations.

Orthonormal systems extend to infinite-dimensional spaces, such as Hilbert spaces, where they enable series expansions, Parseval’s identity, and efficient representations in Fourier or wavelet bases.