Orthonormalität

Orthonormalität is a mathematical property that describes a set of vectors in an inner product space. Specifically, a set of vectors is said to be orthonormal if each vector in the set has a norm (length) of one and is orthogonal (perpendicular) to every other vector in the set. This property is particularly useful in various fields such as linear algebra, signal processing, and quantum mechanics.

In a more formal sense, let V be a vector space equipped with an inner product <, ., >. A set of vectors {v1, v2, ..., vn} in V is orthonormal if:

1. The norm of each vector is one: <vi, vi> = 1 for all i.

2. The vectors are mutually orthogonal: <vi, vj> = 0 for all i ≠ j.

Orthonormal sets are often used to form orthonormal bases, which provide a convenient way to express any vector in the space as a linear combination of the basis vectors. The Gram-Schmidt process is a common method for constructing an orthonormal basis from a given set of linearly independent vectors.

Orthonormality plays a crucial role in the theory of Fourier series and transforms, where it allows for the efficient representation and analysis of signals. In quantum mechanics, orthonormal bases are used to describe the state space of quantum systems, with orthonormal states representing distinct, non-overlapping possibilities.