Inversion error, condition number, and approximate inverses of uncertain matrices

Abstract

The classical condition number is a very rough measure of the effect of perturbations on the inverse of a square matrix. First, it assumes that the perturbation is infinitesimally small. Second, it does not take into account the perturbation structure (e.g., Vandermonde). Similarly, the classical notion of the inverse of a matrix neglects the possibility of large, structured perturbations. We define a new quantity, the structured maximal inversion error, that takes into account both structure and non-necessarily small perturbation size. When the perturbation is infinitesimal, we obtain a “structured condition number”. We introduce the notion of approximate inverse, as a matrix that best approximates the inverse of a matrix with structured perturbations, when the perturbation varies in a given range. For a wide class of perturbation structures, we show how to use (convex) semidefinite programming to compute bounds on the structured maximal inversion error and structured condition number, and compute an approximate inverse. The results are exact when the perturbation is “unstructured”—we then obtain an analytic expression for the approximate inverse. When the perturbation is unstructured and additive, we recover the classical condition number; the approximate inverse is the operator related to the Total Least Squares (orthogonal regression) problem.