Ortogonala

Ortogonala, or orthogonality, is a concept in mathematics that describes a zero inner product relation between objects such as vectors, functions, or subspaces. In real inner product spaces, two nonzero vectors u and v are orthogonal if their inner product ⟨u, v⟩ equals zero; in Euclidean space this corresponds to perpendicularity, since the dot product u·v = 0. A set of nonzero vectors is orthogonal if every distinct pair is orthogonal; if, in addition, each vector has unit length, the set is orthonormal.

In matrix form, the columns of a matrix A are orthogonal if A^T A is diagonal. If the columns are also of unit length, A is an orthogonal matrix, meaning A^T A = I. In complex spaces, the inner product uses conjugation, so ⟨u, v⟩ = ∑ u_i overline{v_i}; orthogonality is defined as ⟨u, v⟩ = 0.

Orthogonality is central to many constructions and analyses. The Gram–Schmidt process converts a linearly independent set into an orthogonal (or orthonormal) set, producing convenient bases for subspaces. In applications, orthogonal families simplify projections and decompositions, as seen in the Pythagorean theorem for orthogonal components and in spectral methods.

Examples include the standard basis vectors in R^n, which are mutually orthogonal; sine and cosine functions over a full period form an orthogonal set in Fourier analysis; and principal components are orthogonal modes that capture maximal variance in data.

Variants of orthogonality arise with different bilinear forms and in broader contexts such as functional analysis, where functions in L^2 are orthogonal if their integral of the product is zero, and in geometry, where orthogonal subspaces intersect at right angles.