Perturbation Bounds for the $QR$ Factorization of a Matrix

Let A be an $m \times n$ matrix of rank n. The $QR$ factorization of A decomposes A into the product of an $m \times n$ 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 $m \times n$ 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 $R^T R$ of a lower triangular matrix and its transpose.