Matrix Theory. From Generalized Inverses to Jordan Form
This book is designed for a “second” course in linear algebra and matrix theory taught at the senior undergraduate and early postgraduate level. It presupposes that the reader has already taken a one-semester course on the elements of linear algebra. The necessary prerequisites are summarized in four appendices. The text is divided into twelve chapters. Chapter 1 is on solutions of systems of linear equations with an emphasis on invertible matrices, and it contains a treatment of the Henderson-Searle formula for the inverse of a sum of matrices and its generalizations. Chapter 2 introduces LU factorization and the Frame algorithm for computing the coefficients of the characteristic polynomial leading to the Cayley-Hamilton theorem. Chapter 3 is on Sylvester’s rank formula and its many consequences. The chapter culminates with the characterization of nilpotent matrices. Left and right inverses are also introduced.
Chapter 4 introduces the main theme of the book: the Moore-Penrose inverse. It is followed in chapter 5 by generalized inverses. A short chapter 6 is about norms followed by chapter 7 on inner products, in particular on the QR factorization and algorithms to find it. The minimum norm and the least square solutions and its connection to the Moore-Penrose inverse are also presented. Chapter 8 discusses orthogonal projections and a connection between the Moore-Penrose inverse and the orthogonal projections on the fundamental subspaces of a matrix. Chapter 9 covers the spectral theorem and chapter 10 covers the primary decomposition theorem, Schur’s triangularization theorem and singular value decomposition. The book then culminates with the Jordan canonical form theorem in chapter 11 and a brief introduction to multilinear algebra in chapter 12. The book contains numerous exercises and homework problems as well as suggestions for further reading. Many concepts are demonstrated with the help of MATLAB.