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Advanced Linear and Matrix Algebra

  • Textbook
  • © 2021

Overview

Authors:
  1. Nathaniel Johnston

    1. Department of Mathematics and Computer Science, Mount Allison University, Sackville, Canada

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  • Motivates a deeper understanding of the abstract structures needed to tackle questions in mathematics, data analysis, and quantum information theory

  • Engages readers with a visual approach that uses color to enhance both content and learning

  • Features a wide selection of theoretical and applied topics to complement the core material

  • Incorporates exercises of all levels

  • Includes supplementary material: sn.pub/extras

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Table of contents (3 chapters)

  1. Front Matter

    Pages i-xvi
    Download chapter PDF

  2. Vector Spaces

    • Nathaniel Johnston
    Pages 1-166

  3. Matrix Decompositions

    • Nathaniel Johnston
    Pages 167-295

  4. Tensors and Multilinearity

    • Nathaniel Johnston
    Pages 297-414

  5. Back Matter

    Pages 415-494
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Keywords

About this book

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. Building on a first course in linear algebra, this book offers readers a deeper understanding of abstract structures, matrix decompositions, multilinearity, and tensors. Concepts draw on concrete examples throughout, offering accessible pathways to advanced techniques.

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. In each case, the author ties the new technique back to familiar ones, to create a coherent set of tools. 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.

Advanced Linear and Matrix Algebra offers students of mathematics, data analysis, and beyond the essential tools and concepts needed for further study. The engaging color presentation and frequent marginal notes showcase the author’s visual approach. A first course in proof-based linear algebra is assumed. An ideal preparation can be found in the author’s companion volume, Introduction to Linear and Matrix Algebra.


Reviews

“The book is well-organized. The main notions and results are well-presented, followed by a discussion and problems with detailed solutions. There are many helpful notes and examples. At the end of each section, the reader can frequently find several computational, true/false, or proof exercises. … There are several illustrative and colorful figures. For instance, those illustrating the examples and remarks about the Gershgorin disc theorem or about the geometric interpretation of the positive semidefiniteness are really helpful.” (Carlos M. da Fonseca, zbMATH 1471.15001, 2021)

Authors and Affiliations

  • Department of Mathematics and Computer Science, Mount Allison University, Sackville, Canada

    Nathaniel Johnston

About the author

Nathaniel Johnston is an Associate Professor of Mathematics at Mount Allison University in New Brunswick, Canada. His research makes use of linear algebra, matrix analysis, and convex optimization to tackle questions related to the theory of quantum entanglement. His companion volume, Introduction to Linear and Matrix Algebra, is also published by Springer.

Bibliographic Information

  • Book Title: Advanced Linear and Matrix Algebra

  • Authors: Nathaniel Johnston

  • DOI: https://doi.org/10.1007/978-3-030-52815-7

  • Publisher: Springer Cham

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: Springer Nature Switzerland AG 2021

  • Hardcover ISBN: 978-3-030-52814-0Published: 20 May 2021

  • Softcover ISBN: 978-3-030-52817-1Published: 21 May 2022

  • eBook ISBN: 978-3-030-52815-7Published: 19 May 2021

  • Edition Number: 1

  • Number of Pages: XVI, 494

  • Number of Illustrations: 15 b/w illustrations, 108 illustrations in colour

  • Topics: Linear and Multilinear Algebras, Matrix Theory, Linear Algebra

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