1. The Moore—Penrose or Generalized Inverse

1. Basic definitions Equations of the form Ax=b,A∈ ℂm×n ,x∈ ℂn ,b∈ ℂm 1 occur in many pure and applied problems. If and is invertible, then the system of equations (1) is, in principle, easy to solve. The unique solution is . If A is an arbitrary matrix in , then it becomes more difficult to solve (1). There may be none, one, or an infinite number of solutions depending on whether b∊R(A) and whether n-rank . One would like to be able to find a matrix (or matrices) C, such that solutions of (1) are of the form Cb. But if b∉R(A), then (1) has no solution. This will eventually require us to modify our concept of what a solution of (1) is. However, as the applications will illustrate, this is not as unnatural as it sounds. But for now we retain the standard definition of solution. To motivate our first definition of the generalized inverse, consider the functional equation y=ƒ (x),x∈S⊆ℝ, 2 where f is a real-valued function with domain . One procedure for solving (2) is to restrict the domain of f to a smaller set so that is one to one. Then an inverse function from to is defined by if and . Thus is a solution of (2) for . This is how the arcsec, arcsin, and other inverse functions are normally defined. The same procedure can be used in trying to solve equation (1). As usual, we let be the linear function from into defined by for . To make a one to one linear transformation it must be restricted to a subspace complementary to N(A). An obvious one is . This suggests the following definition of the generalized inverse.