Numerical Linear Algebra with Applications

Chapter 14

Gram-Schmidt Orthonormalization

Abstract

This chapter develops the Gram-Schmidt process that takes a set of k linearly independent vectors and returns an orthonormal set of k vectors that spans the same subspace. The algorithm begins with the first vector, normalizes it, and then determines the remaining orthonormal sequence by successively subtracting from the next vector in the original sequence the projections of it onto the already computed orthonormal sequence. The difference is orthogonal to all the previously generated vectors, and the algorithm normalize it. There are two versions of the process, classical Gram-Schmidt (CGS) and modified Gram-Schmidt (MGS). During the execution of CGS, the generated vectors are often not ...