Perturbation Bounds for the QR Factorization of a Matrix

Let A be an m n matrix of rank n. The QR factorization of A decomposes A into the product of an m n matrix Q with orthonormal columns and a nonsingular upper triangular matrix R. The decomposition is essentially unique, Q being determined up to the signs of its columns and R up to the signs of its rows. If E is an m n matrix such that A + E is of rank n, then A + E has an essentially unique factorization (Q + W) (R + F). In this paper bounds on \| W \| and \| F \| in terms of \| E \| are given. In addition perturbation bounds are given for the closely related Cholesky factorization of a positive definite matrix B into the product RT R of a lower triangular matrix and its transpose.