Elements of the Theory of Generalized Inverses of Matrices

The purpose of this monograph is to provide a concise introduction to the theory of generalized inverses of matrices that is accessible to undergraduate mathematics majors. Although results from this active area of research have appeared in a number of excellent graduate level text­ books since 1971, material for use at the undergraduate level remains fragmented. The basic ideas are so fundamental, however, that they can be used to unify various topics that an undergraduate has seen but perhaps not related. Material in this monograph was first assembled by the author as lecture notes for the senior seminar in mathematics at the University of Tennessee. In this seminar one meeting per week was for a lecture on the subject matter, and another meeting was to permit students to present solutions to exercises. Two major problems were encountered the first quarter the seminar was given. These were that some of the students had had only the required one-quarter course in matrix theory and were not sufficiently familiar with eigenvalues, eigenvectors and related concepts, and that many -v- of the exercises required fortitude. At the suggestion of the UMAP Editor, the approach in the present monograph is (1) to develop the material in terms of full rank factoriza­ tions and to relegate all discussions using eigenvalues and eigenvectors to exercises, and (2) to include an appendix of hints for exercises.

Contenu

1. Introduction.- 1.1 Preliminary Remarks.- 1.2 Matrix Notation and Terminology.- 1.3 A Rationale for Generalized Inverses.- 2. Systems of Equations and the Moore-Penrose Inverse of a Matrix.- 2.1 Zero, One or Many Solutions of Ax = b.- 2.2 Full Rank Factorizations and the Moore-Penrose Inverse.- 2.3 Some Geometric Illustrations.- 2.4 Miscellaneous Exercises.- 3. More on Moore-Penrose Inverses.- 3.1 Basic Properties of A+.- 3.2 Applications with Matrices of Special Structure.- 3.3 Miscellaneous Exercises.- 4. Drazin Inverses.- 4.1 The Drazin Inverse of a Square Matrix.- 4.2 An Extension to Rectangular Matrices.- 4.3 Expressions Relating Ad and A+.- 4.4 Miscellaneous Exercises.- 5. Other Generalized Inverses.- 5.1 Inverses That Are Not Unique.- Appendix 1: Hints for Certain Exercises.- Appendix 2: Selected References.- Index to Principal Definitions.