Ortonormalis

Ortonormalis is a term used in linear algebra to describe a set of vectors that are pairwise orthogonal and of unit length, commonly referred to as an orthonormal set. When such a set spans a subspace, it is called an orthonormal basis for that subspace; if it spans the entire space, it is an orthonormal basis for the space itself.

In an inner product space V over the real or complex field, with inner product ⟨·,·⟩, a set {v1, ..., vk} is orthonormal if ⟨vi, vj⟩ = 0 for i ≠ j and ⟨vi, vi⟩ = 1 for all i. The span of these vectors is a k-dimensional subspace, and the vectors provide a convenient coordinate system for that subspace.

The Gram matrix G, with entries Gij = ⟨vi, vj⟩, equals the identity matrix for an orthonormal set. If Q is the matrix whose columns are the vectors v1, ..., vk, then Q* Q = I, making Q orthogonal in the real case or unitary in the complex case. Such matrices preserve norms and inner products.

Gram-Schmidt orthonormalization is a process to convert any linearly independent set into an orthonormal set spanning the same subspace.

Examples include the standard basis in R^3, e1 = (1,0,0), e2 = (0,1,0), e3 = (0,0,1), which is orthonormal. More generally, any set obtained by normalizing and orthogonalizing vectors can form an orthonormal basis.

Applications span numerical linear algebra, including QR decomposition, SVD, PCA, and quantum mechanics, where orthonormal bases simplify projections and state representations. The term orthonormalis reflects a Latinized form used in some languages.