Advanced Linear and Matrix Algebra

This textbook emphasizes the interplay between algebra and geometry to motivate the study of advanced linear algebra techniques. Matrices and linear transformations are presented as two sides of the same coin, with their connection motivating inquiry throughout the book. Beginning with a study of vector spaces that includes coordinates, isomorphisms, orthogonality, and projections, the book goes on to focus on matrix decompositions. Numerous decompositions are explored, including the Shur, spectral, singular value, and Jordan decompositions. Tensors and multilinearity complete the book, with a study of the Kronecker product, multilinear transformations, and tensor products. Throughout, "Extra Topic" sections augment the core content with a wide range of ideas and applications, from the QR and Cholesky decompositions to matrix-valued linear maps and semidefinite programming. Exercises of all levels accompany each section.

Format: Paperback / softback
Length: 494 pages
Publication date: 21 May 2022
Publisher: Springer Nature Switzerland AG

This comprehensive textbook delves into the intricate relationship between algebra and geometry, serving as a powerful motivator for advancing the study of advanced linear algebra techniques. Matrices and linear transformations are presented as two interconnected facets, their connection driving the inquiry throughout the book. Building upon a foundational course in linear algebra, this text provides readers with a deeper comprehension of abstract structures, matrix decompositions, multilinearity, and tensors. Concepts are illustrated through concrete examples, offering accessible pathways to advanced methodologies.

The journey begins with a comprehensive exploration of vector spaces, encompassing coordinates, isomorphisms, orthogonality, and projections. This foundation is then expanded upon to focus on matrix decompositions, a central theme of the text. Numerous decompositions are explored, including the Shur, spectral, singular value, and Jordan decompositions. In each instance, the author skillfully connects the new technique to familiar ones, forming a coherent set of tools.

Tensors and multilinearity complete the book, with a thorough study of the Kronecker product, multilinear transformations, and tensor products. Throughout, "Extra Topic" sections enrich the core content with a diverse range of ideas and applications, ranging from the QR and Cholesky decompositions to matrix-valued linear maps and semidefinite programming. Accompanying each section are a multitude of exercises, catering to students of varying levels of expertise.

Advanced Linear and Matrix Algebra serves as a vital resource for students of mathematics, data analysis, and related fields. The engaging color presentation and frequent marginal notes highlight the author's visual approach, making the material accessible and engaging. A prior knowledge of proof-based linear algebra is assumed, and the author's companion volume, Introduction to Linear and Matrix Algebra, serves as an ideal preparation for this comprehensive textbook.

Weight: 944g
Dimension: 177 x 252 x 35 (mm)
ISBN-13: 9783030528171
Edition number: 1st ed. 2021