Matrix Mathematics
A Second Course in Linear Algebra

A modern matrix-based approach to a rigorous second course in linear algebra for mathematics, data science, and physical science majors.

Stephan Ramon Garcia (Author), Roger A. Horn (Author)

9781108837101, Cambridge University Press

Hardback, published 25 May 2023

500 pages
25.9 x 18.3 x 2.8 cm, 1.12 kg

'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

Using a modern matrix-based approach, this rigorous second course in linear algebra helps upper-level undergraduates in mathematics, data science, and the physical sciences transition from basic theory to advanced topics and applications. Its clarity of exposition together with many illustrations, 900+ exercises, and 350 conceptual and numerical examples aid the student's understanding. Concise chapters promote a focused progression through essential ideas. Topics are derived and discussed in detail, including the singular value decomposition, Jordan canonical form, spectral theorem, QR factorization, normal matrices, Hermitian matrices, and positive definite matrices. Each chapter ends with a bullet list summarizing important concepts. New to this edition are chapters on matrix norms and positive matrices, many new sections on topics including interpolation and LU factorization, 300+ more problems, many new examples, and color-enhanced figures. Prerequisites include a first course in linear algebra and basic calculus sequence. Instructor's resources are available.

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.

Subject Areas: Algebra [PBF]