orthogonalization

Orthogonalization is the procedure of converting a set of vectors into an orthogonal (or orthonormal) set that spans the same subspace, with respect to a given inner product. It is a fundamental tool in linear algebra for simplifying computations, projections, and decompositions.

The most common method is the Gram–Schmidt process. Given vectors v1, v2, ..., vk in an inner product space, construct an orthogonal sequence w1, w2, ..., wk by

w1 = v1,

wi = vi − sum_{j=1}^{i−1} (⟨vi, wj⟂ / ⟨wj, wj⟩) wj for i ≥ 2.

If desired, normalize to obtain an orthonormal set ei = wi / ||wi||, yielding ⟨ei, ej⟩ = δij.

Orthogonalization is not limited to real vectors; it applies to complex spaces with the standard inner product, and to function spaces where the inner product is an integral or sum. In finite-dimensional spaces, orthogonal bases simplify many problems, such as projections, least-squares solutions, and transforming matrices to simpler forms.

Variants and related methods include modified Gram–Schmidt, which improves numerical stability; and algorithms that construct an orthonormal basis via QR decomposition, where A = QR with Q having orthonormal columns and R upper triangular. Householder reflections provide another stable approach to produce orthonormal bases.

Properties and applications: orthogonality means zero inner product between distinct basis vectors; orthonormal bases simplify coordinate representations and computations of projections. In practice, orthogonalization aids in solving linear systems, least-squares problems, and in algorithms for eigenvalue problems, signal processing, and data analysis where orthogonality reduces redundancy and improves numerical performance.