TABLE OF CONTENTS Any orthogonal matrix can be written as the product of reflector matrices. Thus the class of reflections is rich enough for all occasions and yet each member is characterized by a single vector which serves to describe its mirror.
- BERESFORD N. PARLETT, The Symmetric Eigenvalue Problem (1998)
A key observation for understanding the numerical properties of the modified Gram-Schmidt algorithm is that it can be interpreted as Householder QR factorization applied to the matrix A augmented with a square matrix of zero elements on top. These two algorithms are not only mathematically ... but also numerically equivalent. This key observation, apparently by Charles Sheffield, was relayed to the author in 1968 by Gene Golub.
- AKE BJ RCK, Numerics of Gram-Schmidt Orthogonalization (1994)
The great stability of unitary transformations in numerical analysis springs from the fact that both the l 2-norm and the Frobenius norm are unitarily invariant. This means in practice that even when rounding errors are made, no substantial growth takes place in the norms of the successive transformed matrices.
- J. H. WILKINSON,
Error Analysis of Transformations Based on the Use of Matrices of the Form I ? 2 ww H (1965)
The QR factorization is a versatile computational tool that finds use in linear equation, least squares and eigenvalue problems. It can be computed in three main ways. The Gram-Schmidt process, which sequentially orthogonalizes the columns of A, is the oldest method and is described in most linear algebra...
