Orthogonális

Orthogonality is a fundamental concept in mathematics describing right angles and independence in inner product spaces. In Euclidean space, two vectors u and v are orthogonal if their inner product u·v equals zero, which means the angle between them is 90 degrees. The concept extends to arbitrary inner product spaces: vectors are orthogonal when their inner product is zero.

A set of nonzero vectors is orthogonal if every pair is orthogonal. If, in addition, each vector has unit length, the set is orthonormal. Orthonormal bases simplify many computations because projections of vectors onto basis elements are immediate, and lengths and angles are preserved by the basis.

An important related object is the orthogonal matrix. An n×n matrix Q is orthogonal when Q^T Q = I. Its columns (and rows) form an orthonormal set; orthogonal matrices preserve inner products and norms, and their inverse equals their transpose. The determinant of an orthogonal matrix is either +1 or -1.

Orthogonality also applies to functions and signals. With respect to an inner product ⟨f,g⟩ = ∫ f(x)g(x) w(x) dx on a domain, two functions are orthogonal if this integral is zero. This underpins Fourier analysis, where sine and cosine functions are mutually orthogonal, and the theory of orthogonal polynomials.

Applications span numerical methods, signal processing, statistics, and data analysis. Orthogonality enables efficient decomposition, noise reduction, and independent treatment of components in datasets or models.

---