Perturbation Bounds for Hyperbolic Matrix Factorizations

Several matrix factorizations depend on orthogonal factors, matrices that preserve the Euclidean scalar product. Some of these factorizations can be extended and generalized to (J, J̃)-orthogonal factors, that is, matrices that satisfy H JH = J̃ , where J and J̃ are diagonal with diagonal elements ±1. The purpose of this work is to analyze the perturbation of matrix factorizations that have a (J, J̃)-orthogonal or orthogonal factor and to give first order perturbation bounds. For each factorization analyzed, we give the sharpest possible first order bound, which yields a condition number. The cost of computing these condition numbers is high. It is usually equivalent to computing the 2-norm of an n2×n2 matrix for a problem of size n. Thus, we also propose less sharp bounds that are less expensive to compute.