Gram-Schmidt (GS) Algorithm

The GS algorithm, formally known as the Gram-Schmidt algorithm, is a fundamental method for orthonormalizing a set of vectors in an inner product space. Developed independently by Jørgen Pedersen Gram and Erhard Schmidt in the early 20th century, this algorithm sequentially processes vectors to create an orthogonal set, then normalizes them to unit length. In computational implementations, the algorithm typically iterates through each vector, subtracting its projections onto all previously orthogonalized vectors. Key operations involve dot products for projection calculations and vector normalization using Euclidean norms. The algorithm finds extensive applications in linear algebra and numerical analysis, with practical implementations in computer graphics (for basis transformations), signal processing (orthogonal filters), and quantum mechanics (state vector orthogonalization). Common numerical considerations include handling linear dependence through modified versions like Modified Gram-Schmidt, which improves numerical stability by reorthogonalizing vectors against accumulated rounding errors.