10. Continuity of the Generalized Inverse
Society for Industrial and Applied Mathematics eBooks · 2009
1. Introduction Consider the following statement: (A) If is a sequence of matrices and converges to an invertible matrix A, then for large enough j, is invertible and converges to . In addition to its obvious theoretical interest, statement (A) has practical computational content. First, if we have a sequence of ‘nice’ matrices which gets close to A, it tells us that gets close to . Thus approximation methods might be of use in computing . Secondly, statement (A) gives us information on how sensitive the inverse of A is to ‘errors’ in determining A. It tells us that if our error in determining A was ‘small’, then the error resulting in due to the error in A will also be ‘small’. This chapter will determine to what extent statement (A) is true for the Moore—Penrose and Drazin generalized inverses. But first, we must discuss what we mean by ‘near’, ‘small’, and ‘converges to’. 2. Matrix norms In linear algebra the most common way of telling when things are close is by the use of norms. Norms are to vectors what absolute value is to numbers. Definition 10.2.1. A function ρ sending a vector space V into the positive reals is called a norm if for all u, v∊V and α∊ℂ; (i) . (ii) , and (iii) (triangle inequality). We will usually denote ρ(u) by ∥ u ∥.