Matrix Mathematics: Theory, Facts, and Formulas: Second Edition
This second edition of Matrix Mathematics represents a major expansion of the original work. While the total number of pages is increased 46% from 752 to 1100, the increase is actually greater since this edition is typeset in a smaller font to facilitate a manageable physical size.
The second edition expands on the first edition in several ways. For example, the new version includes material on graphs (developed within the framework of relations and partially ordered sets), as well as alternative partial orderings of matrices, such as rank subtractivity, star, and generalized L¨owner. This edition also includes additional material on the Kronecker canonical form and matrix pencils; realizations of finite groups; zeros of multi-input, multi-output transfer functions; identities and inequalities for real and complex numbers; bounds on the roots of polynomials; convex functions; and vector and matrix norms.
The additional material as well as works published subsequent to the first edition increased the number of cited works from 820 to 1503, an increase of 83%. To increase the utility of the bibliography, this edition uses the “back reference” feature of LATEX, which indicates where each reference is cited in the text. As in the first edition, the second edition includes an author index. The expansion of the first edition resulted in an increase in the size of the index from 108 pages to 156 pages. The first edition included 57 problems, while the current edition has 73. These problems represent various extensions or generalizations of known results, sometimes motivated by gaps in the literature.
In this edition, I have attempted to correct all errors that appeared in the first edition. As with the first edition, readers are encouraged to contact me about errors or omissions in the current edition, which I will periodically update on my home page.
I am grateful to many individuals who graciously provided useful advice and material for this edition. Some readers alerted me to errors, while others suggested additional material. In other cases I sought out researchers to help me understand the precise nature of interesting results. At the risk of omitting those who were helpful, I am pleased to acknowledge the following: Mark Balas, Jason Bernstein, Vijay Chellaboina, Sever Dragomir, Harry Dym, Masatoshi Fujii, Rishi Graham, Wassim Haddad, Nicholas Higham, Diederich Hinrichsen, Iman Izadi, Pierre Kabamba, Marthe Kassouf, Christopher King, Michael Margliot, Roy Mathias, Peter Mercer, Paul Otanez, Bela Palancz, Harish Palanthandalam-Madapusi, Fotios Paliogiannis, Wei Ren, Mario Santillo, Christoph Schmoeger, Wasin So, Robert Sullivan, Yongge Tian, Panagiotis Tsiotras, G¨otz Trenkler, Chenwei Zhang, and Fuzhen Zhang. As with the first edition, I am especially indebted to my family, who endured three more years of my consistent absence to make this revision a reality. It is clear that any attempt to fully embrace the enormous body of mathematics known as matrix theory is a neverending task. After committing almost two decades to the project, I remain, like Thor, barely able to perceive a dent in the vast knowledge that resides in the hundreds of thousands of pages devoted to this fascinating and incredibly useful subject. Yet, it my hope, that this book will prove to be valuable to all of those who use matrices, and will inspire interest in a mathematical construction whose secrets and mysteries know no bounds.
Dennis S. Bernstein
Ann Arbor, Michigan
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