Standarddotprodukten

Standarddotprodukten refers to the standard dot product on a finite-dimensional vector space, typically real or complex Euclidean space, defined with respect to the fixed standard basis. In real space, for vectors x = (x1, ..., xn) and y = (y1, ..., yn), standarddotprodukten equals x1y1 + x2y2 + ... + xnyn. In complex space, the standard inner product is defined as ⟨x, y⟩ = ∑ x_i conjugate(y_i). In matrix form, this is written as x^T y for real vectors or x^† y (the conjugate transpose) for complex vectors.

The standarddotprodukten induces the Euclidean norm, ||x|| = sqrt(⟨x, x⟩). It is bilinear (real case) or sesquilinear (complex case), symmetric in the real setting and Hermitian in the complex setting, and positive-definite. It is invariant under orthogonal transformations in real space and under unitary transformations in complex space: applying an orthogonal or unitary change of basis preserves inner products and norms.

This standard dot product underpins many geometric and analytic constructions, including the measurement of angles between vectors, projections onto subspaces, and the Gram–Schmidt process for orthogonalization. It also underlies least-squares methods, optimization problems, and many machine learning algorithms that rely on Euclidean distance and inner products.

Notes: the term standarddotprodukten is closely tied to the standard basis; using a different (non-orthonormal) basis effectively changes the form of the inner product. See also dot product, inner product, Euclidean space, Gram matrix.