orthonormal

Orthonormal describes a set of vectors in an inner product space in which each vector has unit length and distinct vectors are orthogonal. In real spaces the inner product is the dot product; in complex spaces it is the Hermitian inner product, so ⟨ei, ej⟩ = 0 for i ≠ j and ⟨ei, ei⟩ = 1. A single unit vector is orthonormal with itself, and a set is orthonormal if the stated conditions hold for all pairs.

If a set is orthonormal, projection and coordinates become simple. The projection of a vector v onto the span of {e1, ..., em} is the sum ∑⟨v, ei⟩ ei. If the set is a basis for the space, this provides the unique expansion v = ∑⟨v, ei⟩ ei. Any linearly independent set can be transformed into an orthonormal set by the Gram–Schmidt process.

In matrix terms, a matrix with orthonormal columns is orthogonal (real case) or unitary (complex case), satisfying Q^T Q = I or Q^H Q = I. Such matrices preserve inner products and norms, making change of basis and spectral computations especially tractable. In a (finite or infinite) Hilbert space, an orthonormal set that is complete is called an orthonormal basis; every vector is the (possibly infinite) sum of its projections onto the basis vectors.

Examples include the standard basis in R^n, which is orthonormal, and sets of sine and cosine functions that form an orthonormal set under a suitable inner product on an interval. Orthonormality underpins many applications in linear algebra, signal processing, and physics, due to its simplifying effect on computations and representations.