Scalar, Vector, and Matrix Mathematics: Theory, Facts, and Formulas
This third edition of Matrix Mathematics, retitled Scalar, Vector, and Matrix Mathematics, is the culmination of seven years of effort to expand the scope of the second edition of this work. In contrast to the first two editions, which were typeset in Computer Modern, this edition is typeset in New Times Roman, which, as a more compact font, accommodates more material per line. The resulting horizontal compression along with tighter line spacing facilitates one of the goals of this edition, which is to substantially expand the scope of the work to include more scalar and vector mathematics than was envisioned in the original “matrix” area. To this end, this edition includes extensive material on scalar inequalities, graphs, groups, geometry, combinatorics, number theory, finite sums and products, special functions, series, and integrals. As an indication of this augmented scope, the second edition of Matrix Mathematics cited 1540 references, whereas the present volume cites 3024.
After three editions and almost three decades of organizational and lexicographical labor, it is perhaps fitting to reflect on mathematics as an artistic and utilitarian endeavor. Mathematics is about the creation of concepts, ideas, and idealizations that are natural, attractive, interesting, and powerful. These attributes may or may not be motivated by physical applications, but what is natural, attractive, interesting, and powerful often turns out to be useful, and vice versa. Mathematical ideas can be appreciated for either their inner beauty or their usefulness. In science and engineering, idealizations serve as approximations; applying and interpreting these approximations is an art guided by analysis, experience, and insight. The ability to think about realworld phenomena in terms of idealizations is essential to the ability of scientists and engineers to interpret and analyze data. The “gap” between idealizations and reality is the space inhabited by mathematically oriented scientists and engineers, who must reconcile the tangibility of data and reality with the ethereal nature of ideas and concepts.
Beyond approximation, mathematics is about characterization, classification, and connection. Characterization is the elucidation of properties possessed by an object such as a number or a function; classification is the construction of taxonomies and hierarchies of concepts and structures; and connections between concepts and constructions reveal deeper and hidden properties. Among the most surprising results are connections that link seemingly unrelated objects in elegant and unexpected ways. e ȷ = 1 is the classic example, but is only one of many that can be found in this book. Each serendipitous connection reinforces the belief that mathematical objects have an independent existence that transcends the ink on the page and the thoughts in our heads. Mathematics is also about composition and decomposition—putting things together and taking things apart. How can a matrix of one type be factored into matrices of another type? How can matrices of one type be combined to form a matrix of another type? Through its focus on conceptual atoms and their agglomeration into complex molecules, mathematics is the chemistry of ideas. Mathematics strives for abstraction. However, the original motivation for this book was to minimize abstraction so that a user of mathematics could find a much-needed result in a sufficiently concrete form to facilitate its correct application. To achieve concreteness, this book avoids ab- stract structures such as fields and vector spaces. But abstractions are powerful. A result proved for topological/metric/normed/inner-product/symplectic spaces is valid for all such spaces, whether they are defined in terms of scalars, vectors, matrices, or functions in finite or (with suitable restric- tions) infinite dimensions. Abstraction provides efficiency of effort, unity of thought, and depth of understanding. With a few exceptions, however, this book intentionally avoids abstraction in order to facilitate the accessibility of the material. But the price paid for this accessibility is to some ex- tent a tunnel view of the larger picture of common ideas and structures. The perfect “handbook of mathematics” would embrace just enough abstraction to unify a huge body of results, and then sys- tematically specialize those results to a multitude of accessible cases that mathematically oriented scientists and engineers might find useful. The perfect handbook remains to be written. Why devote three decades to writing a book such as this one? The main goal is to provide a convenient resource for users of mathematics. This collection complements existing compendia, with coverage and organization that are unique and, hopefully, useful. An unexpected benefit of collecting and organizing this diverse material is the connections that are uncovered. In addition to connections, this collection reveals gaps in knowledge left for researchers to fill and explore. The inclusion of conjectures and problems is a reminder that much remains to be done. As in all worthwhile endeavors, we have finally reached the beginning.
A published review of the second edition of this book aptly suggested that a work of this scope would ideally be the labor of a team of authors. In fact, I have relied heavily on advice and input from many individuals, with 33 acknowledged in the first edition and 44 in the second. For this edition I am indebted to numerous individuals who answered my queries about their books and papers, provided feedback on portions of the manuscript, and contacted me with valuable sugges- tions. These include Shoshana Abramovich, Khaled Aljanaideh, Ovidiu Bagdasar, Oskar Baksalary, Ravindra Bapat, Sanjay Bhat, JC Bourin, Ryan Caverly, Naveen Crasta, Marco Cuturi, Anton de Ruiter, Ayhan Dil, Justin Edmondson, James Forbes, Daniel Franco, Ovidiu Furdui, Michael Gil, Chris Gilbreth, Ankit Goel,Wassim Haddad, Nicholas Higham, Jesse Hoagg, Matthew Holzel, Qing Hui, Jeffrey Humphery, Gidado-Yisa Immanuel, Fuad Kittaneh, Omran Kouba, Peter Larcombe, Minghua Lin, Zongli Lin, Florian Luca, Victor Moll, Robert Piziak, Olivier Ramare, Ranjan Roy, Sneha Sanjeevini, Meiyue Shao, Joseph Silverman, Alina Sintamarian, Wasin So, Valeriu Soltan, Yongge Tian, G¨otz Trenkler, Antai Xie, Doron Zeilberger, Fuzhen Zhang, Xuan Zhou, and Limin Zou. I am especially indebted to Oskar Baksalary, Omran Kouba, Minghua Lin, G¨otz Trenkler, and Yongge Tian for their substantial advice, encouragement, and assistance. Finally, I take full responsibility for the inevitable errors. I encourage you to inform me of any that you may find, and I will post them on my website.
Dennis S. Bernstein
Ann Arbor, Michigan
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