Mathai and Hans J. Haubold Linear Algebra De Gruyter Textbook Unauthenticated Download Date | 6/10/18 9:43 PM Also of Interest Probability and Statistics. A Course for Physicists and Engineers Arak M. Haubold, 2017 ISBN 978-3-11-056253-8, e-ISBN (PDF) 978-3-11-056254-5, e-ISBN (EPUB) 978-3-11-056260-6 Advanced Calculus.
Differential Calculus and Stokes’ Theorem Pietro-Luciano Buono, 2016 ISBN 978-3-11-043821-5, e-ISBN (PDF) 978-3-11-043822-2, e-ISBN (EPUB) 978-3-11-042911-4 Probability Theory and Statistical Applications. A Profound Treatise for Self-Study Peter Zörnig, 2016 ISBN 978-3-11-036319-7, e-ISBN (PDF) 978-3-11-040271-1, e-ISBN (EPUB) 978-3-11-040283-4 Complex Analysis. A Functional Analytic Approach Friedrich Haslinger, 2017 ISBN 978-3-11-041723-4, e-ISBN (PDF) 978-3-11-041724-1, e-ISBN (EPUB) 978-3-11-042615-1 Functional Analysis. A Terse Introduction Gerardo Chacón, Humberto Rafeiro, Juan Camilo Vallejo, 2016 ISBN 978-3-11-044191-8, e-ISBN (PDF) 978-3-11-044192-5, e-ISBN (EPUB) 978-3-11-043364-7 Unauthenticated Download Date | 6/10/18 9:43 PM www.
Mathai and Hans J. Haubold Linear Algebra | A Course for Physicists and Engineers Unauthenticated Download Date | 6/10/18 9:43 PM www.com Mathematics Subject Classification 2010 15-01, 15A03, 15A04, 15A05, 15A09, 15A15, 15A16, 15A18, 15A21, 15A63 Authors Prof. Haubold McGill University United Nations Office for Outer Space Affairs Department of Mathematics and Statistics Vienna International Centre 805 Sherbrooke St. Box 500 Montreal, QC H3A 2K6 1400 Vienna Canada Austria mathai@math.com ISBN 978-3-11-056235-4 e-ISBN (PDF) 978-3-11-056250-7 e-ISBN (EPUB) 978-3-11-056259-0 This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.
For details go to http://creativecommons.org/licenses/by-nc-nd/4. Library of Congress Cataloging-in-Publication Data A CIP catalog record for this book has been applied for at the Library of Congress. Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb. Haubold, published by Walter de Gruyter GmbH, Berlin/Boston.
The book is published with open access at www. Typesetting: VTeX UAB, Lithuania Printing and binding: CPI books GmbH, Leck Cover image: Pasieka, Alfred / Science Photo Library ♾ Printed on acid-free paper Printed in Germany www.com Unauthenticated Download Date | 6/10/18 9:43 PM www.com Basic properties of vectors, matrices, determinants, eigenvalues and eigenvectors are discussed. Then, applications of matrices and determinants to various areas of sta- tistical problems such as principal components analysis, model building, regression analysis, canonical correlation analysis, design of experiments etc. Ap- plications of vector/matrix derivatives in the simplification of Taylor expansions of functions of many real scalar variables are considered.
Jacobians of matrix transfor- mations of real-valued scalar functions of matrix argument, maxima/minima prob- lems, optimizations of linear forms, quadratic forms, bilinear forms with linear and quadratic constraints are examined. Matrix sequences and series, convergence of ma- trix series etc. and applications in physical sciences, chemical sciences, social sci- ences, input-analysis, linear programming problem, non-linear least squares and dy- namic programming problems etc. are studied in this book.
Each topic is motivated by real-life situations and each concept is illustrated with examples and counter examples. The book is class-tested since 1999. It is written with the experience of teaching fifty years in various universities around the world. The first three Modules of the Centre for Mathematical and Statistical Sciences (CMSS)are com- bined to make this book.
These Modules are used for intensive undergraduate mathe- matics training camps of CMSS. Each camp is a 10-day intensive training course with 40 hours of lectures and 40 hours of problem-solving sessions. Thirty such camps are already conducted by CMSS. Only high school level mathematics is assumed.
The book is written as a self-study material. Each topic is brought from fundamentals to the se- nior undergraduate to graduate level. Usual doubts of the students on various topics are answered in the book. Since 2004, the material in this book was made available to UN-affiliated Re- gional Centres for Space Science and Technology Education, located in India, China, Morocco, Nigeria, Jordan, Brazil, and Mexico (http://www.org/oosa/en/ ourwork/psa/regional-centres/index.
Since 1988 the material was taken into account for the development of educa- tion curricula in the fields of remote sensing and geographic information systems, satellite meteorology and global climate, satellite communications, space and atmo- spheric science, and global navigation satellite systems (http://www.org/ oosa/en/ourwork/psa/regional-centres/study_curricula. As such the material was considered to be a prerequisite for applications, teach- ing, and research in space science and technology. It was also a prerequisite for the nine-months post-graduate courses in the five disciplines of space science and tech- nology, offered by the Regional Centres on an annual basis to participants from all 194 Member States of the United Nations. Since 1991, whenever suitable at the research level, the material in this book was utilized in lectures in a series of annual workshops and follow-up projects of the so- Open Access.
Haubold, published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.1515/9783110562507-201 Unauthenticated Download Date | 6/10/18 9:43 PM www.com VI | called Basic Space Science Initiative of the United Nations (http://www.org/ oosa/en/ourwork/psa/bssi/index. As such the material was considered a prerequisite for teaching and research in astronomy and physics. RIPPLE SIGHTING The cosmic dance of two black holes warped spacetime as the pair spiraled inward and merged, creating gravitational waves (illustration below).
Advance Laser Interferometer Gravitational-Wave Observatory (LIGO) detected these ripples, produced by black holes eight and 14 times the mass of the sun, on Decem- ber 26, 2015. Einstein’s theory of general relativity was 100 years old in 2015. It has been very important in applications such as GPS (GNSS), and tremendously successful in understanding astrophysical systems like black holes. Gravitational waves, which are ripples in the fabric of space and time produced by violent events in the distant uni- verse – for example, by the collision of two black holes or by the cores of supernova explosions – were predicted by Albert Einstein in 1916 as a consequence of his general theory of relativity.
Gravitational waves are emitted by accelerating masses much in the same way electromagnetic waves are produced by accelerating charges, such as radio waves radiated by electrons accelerating in antennas. As they travel to Earth, these ripples in the space–time fabric carry information about their violent origins and about the nature of gravity that cannot be obtained by traditional astronomical observations using light. Gravitational waves have now been detected directly. Scien- Unauthenticated Download Date | 6/10/18 9:43 PM www.com | VII tists do, however, have great confidence that they exist because their influence on a binary pulsar system (two neutron stars orbiting each other) has been measured ac- curately and is in excellent agreement with the predictions.
Directly detecting gravi- tational waves has confirmed Einstein’s prediction in a new regime of extreme rela- tivistic conditions, and open a promising new window into some of the most violent and cataclysmic events in the cosmos. The GNSS education curricula provides oppor- tunities to teach navigation and do research in astrophysics (basic space science). The development of the education curricula (illustrated above) started in 1988 at UN Head- quarters in New York, the specific GNSS curriculum emanated only in 1999 after the UNISPACE III Conference, held at and hosted by the United Nations at Vienna. Usually students from other areas, other than mathematics, are intimidated by seeing theorems and proofs.
Hence no such phrase as “theorem” is used in the book. Main results are called “results” and are written in bold so that the material will be user-friendly. This book can be used as a textbook for a beginning undergraduate level course on vectors, matrices and determinants, and their applications, for students from all disciplines. Unauthenticated Download Date | 6/10/18 9:43 PM www.com Unauthenticated Download Date | 6/10/18 9:43 PM www.com Preface The basic material in this book originated from a course given by the first author at the University of Texas at El Paso in 1998–1999 academic year.
Students from math- ematics, engineering, biology, economics, physics and chemistry were in the class. The textbook assigned to the course did not satisfy the students from any of the dis- ciplines, including mathematics. Hence Dr Mathai started developing a course from fundamentals, assuming no background, with lots of examples and counter exam- ples taken from day to day life. All sections of the students enjoyed the course.
Dr Mathai gave courses on calculus and linear algebra and for both of these courses he developed his own materials in close interaction with students. The El Paso experi- ment was initially for one semester only but, due to the popularity, extended to more semesters. During 2000 to 2006 these notes were developed into CMSS Modules and based on these Modules, occasional courses were given for teachers and students at various levels in Kerala, India, as per requests from teachers. From 2007 onward CMSS became a Department of Science and Technology, Government of India centre for mathemat- ical and statistical sciences.
Modules in other areas were also developed during this period, and by 2014, ten Modules were developed. As a Life Member of CMSS, the second author is an active participant of all pro- grams at CMSS, including the undergraduate mathematics training camps, Ph.D train- ing etc. and he is also a frequent visitor to CMSS to participate in and contribute to various activities. Chapter 1 is devoted to all basic properties of vectors as ordered set of real num- bers, Each definition is motivated by real-life examples.
After introducing major prop- erties of vectors with the real elements, vectors in the complex domain are considered and more rigorous definitions are introduced. Chapter 1 ends with Gram–Schmidt or- thogonalization process. Chapter 2 deals with matrices. Again, all definitions and properties are introduced from real-life situations.
Roles of elementary matrices and elementary operations in solving linear equations, checking consistency of linear systems, checking linear de- pendence of vectors, evaluating rank of a matrix, canonical reductions of quadratic and bilinear forms, triangularizations and diagonalizations of matrices, computing inverses of matrices etc. Chapter 3 deals with determinants. An axiomatic definition is introduced. Various types of expansions of determinants are given.
Role of elementary matrices in evalu- ating determinants is highlighted. This chapter melts into Chapter 4 on eigenvalues and eigenvectors and their properties. Chapters 5 and 6 are on applications of matrices and determinants to various disciplines. Applications to maxima/minima problems, constrained maxima/minima, optimization of linear, quadratic and bilinear forms, with linear and quadratic con- Open Access.
Haubold, published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.1515/9783110562507-202 Unauthenticated Download Date | 6/10/18 9:44 PM www.com X | Preface straints are considered. For each optimization, at least one practical procedure such as principal components analysis, canonical correlation analysis, regression analysis etc. Some additional topics are also developed in Chapter 6.
Matrix poly- nomials, matrix sequences and series, convergence, norms of matrices, singular value decomposition of matrices, simultaneous reduction of matrices to diagonal forms etc. are also discussed in Chapter 6. Mathai 14th March 2017 H. Haubold Unauthenticated Download Date | 6/10/18 9:44 PM www.com Acknowledgement Several people have contributed directly or indirectly to make these Modules to the present levels.
The financial support from the Department of Science and Technol- ogy, Government of India (DST), during the period 2007 to 2014 helped in printing and reprinting of the Modules. This helped in improving the quality of the material. Acharya, then Dr A. Singh and then Dr P.
Malhotra of the mathematical sciences division of DST, New Delhi, deserve special mention in providing research funds to CMSS. At the termination of DST support, Dr V. Rajasekharan Pillai, for- mer Executive Vice-President of the Kerala State Council for Science, Technology and Environment (KSCSTE), a man with vision, took steps to support CMSS so that its activ- ities of research, undergraduate mathematics training camps and Ph. training could continue uninterrupted.
Princy of CMSS was kind enough to reset all figures in the current book. She deserves special mention. Ms Sini Devassy, the office manager of CMSS deserves special mention. graduates from CMSS, Dr Seema S.
Nair, Dr Nicy Sebastian, Dr Dhannya P. Joseph, Dr Dilip Kumar, Dr P. Princy, Dr Naiju M. Thomas, Dr Anita Thomas, Dr Shanoja S.
Pai, Dr Ginu Varghese, Dr Sona P.