Perturbation Analysis of Matrix Equations and Decompositions

A matrix computational problem is a function which maps a set of data (usually in the form of a collection of matrices) into a matrix space whose elements are the desired solutions. If a particular data is perturbed then the corresponding solution is also perturbed. The goal of the norm-wise perturbation analysis is to estimate the norm of the perturbation in the solution as a function of the norms of the perturbations in the data. In turn, in the component-wise perturbation analysis the modules of the elements of the solution are estimated as functions of the modules of the perturbations in the data. The perturbation analysis can be local and nonlocal. In the local analysis it is supposed that the perturbations in the data are asymptotically small and a local bound for the perturbation in the solution is constructed which is valid for first order (small) perturbations. A disadvantage of the local analysis is that normally it does not have a priori measure on how ‘small’ the data perturbations must be in order to guarantee the results from the local estimates being correct. A desirable property of the local bounds is formulated as follows: a perturbation bound is asymptotically exact when for some perturbations it is arbitrarily close to the actual perturbed quantity. On the other hand the nonlocal perturbation analysis produces perturbation estimates which are rigorously valid in a certain set of data perturbations. The price of this advantage is that the nonlocal perturbation bounds may be too pessimistic in certain cases and/or the domain of validity of these bounds may be relatively small. However, a desirable property of the nonlocal bounds is that within first order perturbations they coincide with the improved local bounds. In this chapter we consider the basic methods for perturbation analysis of matrix algebraic equations and unitary (orthogonal in particular) matrix decompositions. The nonlocal perturbation analysis of matrix equations includes several steps: (a) reformulation of the perturbed problem as an equivalent operator equation with respect to the perturbation in the solution; (b) construction of a Lyapunov majorant for the corresponding operator; (c) application of fixed point principles in order to prove that the perturbed equation has a solution; (d) estimation of the solution of the associated majorant equation. The latter estimate gives the desired nonlocal perturbation bound. The nonlocal perturbation analysis of unitary matrix decompositions is based on a systematic use of the method of splitting operators and vector Lyapunov majorants. In this way nonlocal perturbation bounds are derived for the basic unitary decompositions of matrices (QR decomposition and Schur decomposition in particular) and for important problems in the theory of linear time-invariant control systems: transformation into unitary canonical form and synthesis of closed-loop systems with desired equivalent form.