Other Generalized Inverses

Abstract

Given matrices A and X, subsets of the relations in (1.1) to (1.5) other than those used to define A⁺ and A_d provide additional types of generalized inverses. Although not unique, some of these generalized inverses exhibit the essential properties of A⁺ required in various applications. For example, observe that only the condition AXA = A was needed to characterize consistent systems of equations Ax = b by the relation AXb = b in (2.4). Moreover, if A and X also satisfy (XA)^H = XA, then XA = A⁺A, by Exercise 4.10, and with A⁺b a particular solution of Ax = b, the general solution in Exercise 2.21 can be written as

$$\text{x =}{{\text{A}}^{+}}\text{b + (I-XA)y}$$

with the orthogonal decomposition

$${{\left\| \text{x} \right\|}^{2}}\text{ = }{{\left\| {{\text{A}}^{+}}\text{b} \right\|}^{2}}\text{ + }{{\left\| (\text{I-XA})\text{y} \right\|}^{2}}.$$

(Note that this is an extension of the special case of matrices with full row rank used in the proof of Theorem 2.) In this section we consider relationships among certain of these generalized inverses in terms of full rank factorizations, and illustrate the construction of such inverses with numerical examples.