Orthornormal

Orthornormal is typically the misspelled form of the term orthonormal, which refers to a specific relationship among vectors in an inner product space. A set of nonzero vectors {e1, e2, ..., en} is orthonormal if the inner product satisfies <ei, ej> = 0 whenever i ≠ j and <ei, ei> = 1 for all i. In particular, each vector has unit length and is orthogonal to every other vector in the set.

In R^n with the standard inner product, the standard basis is orthonormal, and any orthonormal set spans a subspace with the property that the norm of any linear combination equals the square root of the sum of the squares of its coefficients in that basis. An orthonormal basis is an orthonormal set that also spans the space, so every vector has a unique representation as a sum of its inner products with the basis vectors.

Key properties include that the coordinates of a vector x in an orthonormal basis are given by the inner products x_i = <x, ei>, and that ||x||^2 = Σ |<x, ei>|^2. Orthogonal transformations formed from an orthonormal basis preserve inner products and norms.

Gram–Schmidt is a standard method to convert any linearly independent set into an orthonormal set. In infinite-dimensional Hilbert spaces, an orthonormal basis allows expansion x = Σ <x, ei> ei, with Parseval’s identity ensuring ||x||^2 = Σ |<x, ei>|^2 when the basis is complete.

Applications span Fourier analysis, quantum mechanics, signal processing, and principal component analysis, where orthonormal bases facilitate simple representations and energy preservation. The correct term is orthonormal; orthronormal is a common misspelling.