Orthogonalizing

Orthogonalizing is the process of transforming a set of vectors into an orthogonal set, or an orthonormal set if the vectors are also normalized, spanning the same subspace. In an inner product space, two vectors are orthogonal if their inner product is zero; in Euclidean space this is the dot product being zero. The purpose is to simplify computations, such as projections, decompositions, and solving linear systems.

The most common algorithm is the Gram-Schmidt process. Starting with the original vectors, it constructs an orthogonal basis {u1, u2, ..., uk} by setting u1 = v1 and then, for j > 1, forming wj = vj minus the sum of its projections onto the previously computed vectors: wj = vj − sum_{i<j} proj_{ui}(vj). Normalizing wj yields uj = wj / ||wj||, producing an orthonormal set. Classical Gram-Schmidt is straightforward but can be numerically unstable; the modified Gram-Schmidt variant improves stability by reordering and reordering projections during computation.

Alternative methods include Householder reflections and QR factorization, where a matrix A is decomposed as A = Q R with Q having orthonormal columns. These approaches often provide greater numerical robustness and efficiency in practice.

Limitations include numerical rounding errors that can erode orthogonality, sometimes necessitating re-orthogonalization. If the original set is linearly dependent, the orthogonalized set spans the same subspace but contains fewer vectors equal to the rank.

Applications span projections, least squares, eigenvalue computations, and signal processing. In data analysis, orthogonal bases underpin algorithms such as principal component analysis, which seeks an orthogonal basis aligned with maximal variance.