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Matrix Analysis and Applied Linear Algebra

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by Carl D. Meyer


2000 / xii + 718 pages / Hardcover / ISBN: 978-0-898714-54-8 / List Price $110.00 / SIAM Member Price $77.00 / Includes CD-ROM and Solutions Manual / Order Code OT71

"This book combines the best of what you look for in a reference and a textbook. It is comprehensive and detailed but with so many great problems and examples that it is guaranteed to excite the undergraduate reader. I enjoyed the book throughout, but found the treatment of the FFT to be particularly original and effective." -- Charles Van Loan, Professor and Chair, Department of Computer Science, Cornell University.

"Carl Meyer's book is an outstanding addition to the vast literature in this area. Its most distinctive feature is a seamless integration of the theoretical, computational, and applied aspects of the subject, which stems from the author's extensive experience in both teaching and research. The author's clear and elegant expository style is enlivened by a generous sprinkling of historical notes and aptly chosen quotations from famous mathematicians, making this book a delight to read. If this textbook will not succeed in awakening your students' interest in matrices and their uses, nothing else will." -- Michele Benzi, Los Alamos National Laboratory.

"The book covers an impressive range of material. It contains a number of topics not found in similar books. Professor Meyer takes great care in explaining abstract concepts." -- Ilse Ipsen, Professor, North Carolina State University.

"In studying for the final exam, I was able to solidify much of what we'd gone over during the semester by reading over the text and doing the problems. However, this time through, I gained a lot more respect for the textbook. I didn't notice at the time, but the way the earlier problems foreshadowed techniques yet to come really helped me get everything to click into place. I am sincerely impressed with how much I learned." -- Gregory Nusz, Caldwell Scholar, North Carolina State University.

"I will say that I really enjoy the prose. It is a rare combination when the enthusiasm shines through, focused by erudition." -- Cleve Ashcraft, Livermore Software Technology Corporation.

"I like the book: the theory is sound, numerical performance and possible pitfalls of the algorithms are well discussed, and it contains interesting historical remarks." -- Walter Gander, Chairman, Computer Science Department, ETH Zurich.

Matrix Analysis and Applied Linear Algebra is an honest math text that circumvents the traditional definition-theorem-proof format that has bored students in the past. Meyer uses a fresh approach to introduce a variety of problems and examples ranging from the elementary to the challenging and from simple applications to discovery problems. The focus on applications is a big difference between this book and others. Meyer's book is more rigorous and goes into more depth than some. He includes some of the more contemporary topics of applied linear algebra which are not normally found in undergraduate textbooks. Modern concepts and notation are used to introduce the various aspects of linear equations, leading readers easily to numerical computations and applications. The theoretical developments are always accompanied with examples, which are worked out in detail. Each section ends with a large number of carefully chosen exercises from which the students can gain further insight.

The textbook contains more than 240 examples, 650 exercises, historical notes, and comments on numerical performance and some of the possible pitfalls of algorithms. It comes with a solutions manual that includes complete solutions to all of the exercises. As an added bonus, a CD-ROM is included that contains a searchable copy of the entire textbook and all solutions. Detailed information on topics mentioned in examples, references for additional study, thumbnail sketches and photographs of mathematicians, and a history of linear algebra and computing are also on the CD-ROM, which can be used on all platforms.

Students will love the book's clear presentation and informal writing style. The detailed applications are valuable to them in seeing how linear algebra is applied to real-life situations. One of the most interesting aspects of this book, however, is the inclusion of historical information. These personal insights into some of the greatest mathematicians who developed this subject provide a spark for students and make the teaching of this topic more fun.

This text is a traditional linear algebra book designed to prepare students in mathematics, science, and engineering to deal with a broad range of applications. The coverage is very broad so that it will appeal to professors teaching at the junior-senior to beginning graduate level. Because of the broad coverage, it has more flexibility than many texts. There is enough material so that professors can pick and choose topics depending on the level of a particular class, and enough depth so that it can be tailored to many different courses. Meyer begins at an introductory level and progresses to topics more appropriate for a first year graduate course.

Chapter 1: Linear Equations. Introduction; Gaussian Elimination and Matrices; Gauss-Jordan Method; Two-Point Boundary-Value Problems; Making Gaussian Elimination Work; Ill-Conditioned Systems; Chapter 2: Rectangular Systems and Echelon Forms. Row Echelon Form and Rank; The Reduced Row Echelon Form; Consistency of Linear Systems; Homogeneous Systems; Nonhomogeneous Systems; Electrical Circuits; Chapter 3: Matrix Algebra. From Ancient China to Arthur Cayley; Addition, Scalar Multiplication, and Transposition; Linearity; Why Do It This Way?; Matrix Multiplication; Properties of Matrix Multiplication; Matrix Inversion; Inverses of Sums and Sensitivity; Elementary Matrices and Equivalence; The LU Factorization; Chapter 4: Vector Spaces. Spaces and Subspaces; Four Fundamental Subspaces; Linear Independence; Basis and Dimension; More About Rank; Classical Least Squares; Linear Transformations; Change of Basis and Similarity; Invariant Subspaces; Chapter 5: Norms, Inner Products, and Orthogonality. Vector Norms; Matrix Norms; Inner Product Spaces; Orthogonal Vectors; Gram-Schmidt Procedure; Unitary and Orthogonal Matrices; Orthogonal Reduction; The Discrete Fourier Transform; Complementary Subspaces; Range-Nullspace Decomposition; Orthogonal Decomposition; Singular Value Decomposition; Orthogonal Projection; Why Least Squares?; Angles Between Subspaces; Chapter 6: Determinants. Determinants; Additional Properties of Determinants; Chapter 7: Eigenvalues and Eigenvectors. Elementary Properties of Eigensystems; Diagonalization by Similarity Transformations; Functions of Diagonalizable Matrices; Systems of Differential Equations; Normal Matrices; Positive Definite Matrices; Nilpotent Matrices and Jordan Structure; The Jordan Form; Functions of Non-diagonalizable Matrices; Difference Equations, Limits, and Summability; Minimum Polynomials and Krylov Methods; Chapter 8: Perron-Frobenius Theory of Nonnegative Matrices. Introduction; Positive Matrices; Nonnegative Matrices; Stochastic Matrices and Markov Chains.

ISBN: 9780898714548

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