Ortogonalisia

Ortogonalisia is a concept in mathematics describing the mutual orthogonality of vectors, functions, or signals within an inner product space. Two nonzero vectors u and v are orthogonal when their inner product ⟨u, v⟩ equals zero. In real Euclidean space with the standard dot product this coincides with geometric perpendicularity; in complex spaces the inner product is conjugate-symmetric, so ⟨u, v⟩ = 0 still indicates orthogonality.

An orthogonal set has pairwise zero inner products; an orthonormal set is orthogonal and each vector has unit length. If a basis is orthonormal, any vector can be expressed by simple coordinates: v = sum_i ⟨v, u_i⟩ u_i. For a non-orthonormal basis, coordinates generally require solving a linear system or using projection formulas.

Gram-Schmidt is a standard algorithm to construct an orthogonal, and optionally orthonormal, basis from any linearly independent set. This property underpins many numerical methods, including least-squares, QR decomposition, and various eigenvalue algorithms, as well as signal processing techniques where orthogonality reduces interference and simplifies reconstruction. The concept also generalizes to function spaces, such as L2, with orthogonal basis functions like sine and cosine in Fourier series.

Ortogonalisia is used in several languages to refer to the concept of orthogonality; it is closely related to terms such as orthogonality, orthonormality, and inner products.