Orthogonalize

Orthogonalize is the process of converting a set of vectors into an orthogonal set with respect to an inner product. In Euclidean space, two vectors are orthogonal if their dot product is zero. If, in addition, each vector has unit length, the set is orthonormal. Orthogonalization is fundamental in simplifying projections, decompositions, and solutions to linear systems.

The most common method is the Gram-Schmidt process. Given linearly independent vectors v1, v2, ..., vk, the procedure constructs an orthogonal set u1, u2, ..., uk. Start with u1 = v1. For each j from 2 to k, subtract the projections of v_j onto all previous u_i: w_j = v_j − sum_{i=1}^{j−1} (v_j · u_i) / (u_i · u_i) · u_i, and then set u_j = w_j. If an orthonormal set is desired, normalize each u_j by e_j = u_j / ||u_j||. A variant, the modified Gram-Schmidt algorithm, improves numerical stability by reordering and reOrthogonalizing during the process.

Orthogonalization is closely related to QR factorization, where a matrix A with column vectors is decomposed as A = Q R with Q having orthonormal columns and R upper triangular. Householder transformations offer another robust approach to producing orthogonal factors and are widely used in numerical linear algebra.

Applications of orthogonalization include computing orthonormal bases for subspaces, simplifying least squares problems by exploiting Q^T b, stabilizing eigenvalue and singular value computations, and enabling efficient projections and decompositions in signal processing and data analysis. Practical issues include handling dependent vectors, which reduce the attainable number of orthogonal vectors, and mitigating numerical loss of orthogonality due to finite-precision arithmetic, sometimes addressed by reorthogonalization.