orthogonale

Orthogonale is the Italian term for orthogonal and describes a relation of right-angle structure in mathematics and related fields. In general, two elements are orthogonal when their defined inner product equals zero, which generalizes the geometric notion of perpendicularity. In real vector spaces this inner product is typically the dot product; in complex spaces the inner product uses conjugation, so the condition becomes the conjugate-linear form ⟨u, v⟩ = 0.

An orthogonal set consists of nonzero vectors that are pairwise orthogonal. If every vector in the set has unit length, the set is orthonormal. Orthogonality simplifies many calculations: projections onto a subspace can be performed along its orthogonal complement, and decompositions of vectors into orthogonal components are straightforward.

An orthogonal transformation is a linear map that preserves inner products, equivalently preserving lengths and angles. Its matrix Q satisfies QᵀQ = I, and its inverse is Q⁻¹ = Qᵀ. Such transformations represent rotations and reflections and are fundamental in preserving geometric structure under change of basis.

In numerical linear algebra, Gram–Schmidt orthogonalization constructs an orthogonal (or orthonormal) basis from a linearly independent set, enabling stable least-squares solutions and efficient spectral methods. In functional spaces, two functions are orthogonal if their inner product, often an integral like ∫ f(x)g(x) dx over a domain, vanishes. This concept underpins Fourier series, and the theory of orthogonal polynomials such as Legendre and Hermite families.

Applications of orthogonality span many areas: experimental design uses orthogonal contrasts to separate effects cleanly; signal processing employs orthogonal codes and bases; statistics and data analysis rely on orthogonality to simplify models and improve interpretability. The concept extends to any space equipped with an appropriate inner product or sesquilinear form.