Contents; Preface; Notation; 1. Vector Spaces; 2. Bases and Similarity; 3. Block Matrices; 4. Rank, Triangular Factorizations, and Row Equivalence; 5. Inner Products and Norms; 6. Orthonormal Vectors; 7. Unitary Matrices; 8. Orthogonal Complements and Orthogonal Projections; 9. Eigenvalues, Eigenvectors, and Geometric Multiplicity; 10. The Characteristic Polynomial and Algebraic Multiplicity; 11. Unitary Triangularization and Block Diagonalization; 12. The Jordan Form: Existence and Uniqueness; 13. The Jordan Form: Applications; 14. Normal Matrices and the Spectral Theorem; 15. Positive Semidefinite Matrices; 16. The Singular Value and Polar Decompositions; 17. Singular Values and the Spectral Norm; 18. Interlacing and Inertia; 19. Norms and Matrix Norms; 20. Positive and Nonnegative Matrices; References; Index.
A modern matrix-based approach to a rigorous second course in linear algebra for mathematics, data science, and physical science majors.
Stephan Ramon Garcia is W .M. Keck Distinguished Service Professor and Chair of the Department of Mathematics and Statistics at Pomona College. He is the author of five books and over 100 research articles in operator theory, complex analysis, matrix analysis, number theory, discrete geometry, and combinatorics. He has served on the editorial boards of the Proceedings of the American Mathematical Society, Notices of the American Mathematical Society, Involve, and The American Mathematical Monthly. He received six teaching awards from three different institutions and is a fellow of the American Mathematical Society, which has awarded him the inaugural Dolciani Prize for Excellence in Research. Roger A. Horn was Professor and Chair of the Department of Mathematical Sciences at the Johns Hopkins University, and Research Professor of Mathematics at the University of Utah until his retirement in 2015. His publications include Matrix Analysis, 2nd edition (Cambridge, 2012) and Topics in Matrix Analysis (with Charles R. Johnson, Cambridge, 1991), as well as more than 100 research articles in matrix analysis, statistics, health services research, complex variables, probability, differential geometry, and analytic number theory. He was the editor of The American Mathematical Monthly and has served on the editorial boards of the SIAM Journal of Matrix Analysis, Linear Algebra and its Applications, and the Electronic Journal of Linear Algebra.
'A broad coverage of more advanced topics, rich set of exercises,
and thorough index make this stylish book an excellent choice for a
second course in linear algebra.' Nick Higham, University of
Manchester
'This textbook thoroughly covers all the material you'd expect in a
Linear Algebra course plus modern methods and applications. These
include topics like the Fourier transform, eigenvalue adjustments,
stochastic matrices, interlacing, power method and more. With 20
chapters of such material, this text would be great for a
multi-part course and a reference book that all mathematicians
should have.' Deanna Needell, University of California, Los
Angeles
'The original edition of Garcia and Horn's Second Course in Linear
Algebra was well-written, well-organized, and contained several
interesting topics that students should see - but rarely do in
first-semester linear algebra - such as the singular value
decomposition, Gershgorin circles, Cauchy's interlacing theorem,
and Sylvester's inertia theorem. This new edition also has all of
this, together with useful new material on matrix norms. Any
student with the opportunity to take a second course on linear
algebra would be lucky to have this book.' Craig Larson, Virginia
Commonwealth University
'An extremely versatile Linear Algebra textbook that allows
numerous combinations of topics for a traditional course or a more
modern and applications-oriented class. Each chapter contains the
exact amount of information, presented in a very easy-to-read
style, and a plethora of interesting exercises to help the students
deepen their knowledge and understanding of the material.' Maria
Isabel Bueno Cachadina, University of California, Santa Barbara
'This is an excellent textbook. The topics flow nicely from one
chapter to the next and the explanations are very clearly
presented. The material can be used for a good second course in
Linear Algebra by appropriately choosing the chapters to use.
Several options are possible. The breadth of subjects presented
makes this book a valuable resource.' Daniel B. Szyld, Temple
University and President of the International Linear Algebra
Society
'With a careful selection of topics and a deft balance between
theory and applications, the authors have created a perfect
textbook for a second course on Linear Algebra. The exposition is
clear and lively. Rigorous proofs are supplemented by a rich
variety of examples, figures, and problems.' Rajendra Bhatia,
Ashoka University
'The authors have provided a contemporary, methodical, and clear
approach to a broad and comprehensive collection of core topics in
matrix theory. They include a wealth of illustrative examples and
accompanying exercises to re-enforce the concepts in each chapter.
One unique aspect of this book is the inclusion of a large number
of concepts that arise in many interesting applications that do not
typically appear in other books. I expect this text will be a
compelling reference for active researchers and instructors in this
subject area.' Shaun Fallat, University of Regina
'It starts from scratch, but manages to cover an amazing variety of
topics, of which quite a few cannot be found in standard textbooks.
All matrices in the book are over complex numbers, and the
connections to physics, statistics, and engineering are regularly
highlighted. Compared with the first edition, two new chapters and
300 new problems have been added, as well as many new conceptual
examples. Altogether, this is a truly impressive book.' Claus
Scheiderer, University of Konstanz
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