The Gram-Schmidt process is a fundamental algorithm in linear algebra that transforms any basis of a finite-dimensional inner product space into an orthogonal or orthonormal basis through sequential projection subtraction and normalization. This procedure relies on the principle of vector decomposition, where each new basis vector is constructed by subtracting its projections onto previously computed orthogonal vectors from the original input vectors. Consequently, QR factorization utilizes this mechanism to decompose a general matrix A into the product of an orthogonal matrix Q containing orthonormal columns and an upper triangular matrix R, providing critical theoretical utility for solving linear systems, eigenvalue problems, and least squares approximations without requiring direct numerical inversion.
The Gram-Schmidt process is a fundamental algorithm in linear algebra that transforms any basis of a finite-dimensional inner product space into an orthogonal or orthonormal basis through sequential …