Advanced Linear and Matrix Algebra

Advanced Linear and Matrix Algebra coverThis 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 companion volume, Introduction to Linear and Matrix Algebra.

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  • From zbMATH (review by Carlos M. da Fonseca):

    This highly recommendable textbook is the companion volume of [Johnston’s other book].

    . . .

    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.

  • From the paper “Various Course Proposals for: Mathematics with a View Towards (the Theoretical Underpinnings of) Machine Learning” by Marc S. Paolella:

    As the second of two volumes on linear algebra, both intended for undergraduates, with the first having been mentioned and praised in Section 3.2, use of this book would clearly be highly desirable, notably, but not only, if the first book was used in a first course. As mentioned earlier regarding the first book, Johnston’s second book is, similarly, optically fantastic, has excellent graphics, use of color in the text and graphics, contains many interesting topics, all seemingly very well done, as well as useful appendices to get the reader up to speed. Skimming in the second book makes me want to stop what I am doing and read the whole thing. . . . If this book is not chosen as the primary one, it certainly should enter, and be high on, a supplementary reading list.

Citation/bibliographic information:

  • Author: Nathaniel Johnston
  • Cite as: N. Johnston. Advanced Linear and Matrix Algebra. Springer International Publishing, 2021.
  • DOI: 10.1007/978-3-030-52815-7
  • eBook ISBN: 978-3-030-52815-7
  • Hardcover ISBN: 978-3-030-52814-0
  • BibTeX entry: 
    @book{johnston2021advanced,
      title={Advanced Linear and Matrix Algebra},
      author={Nathaniel Johnston},
      isbn={978-3-030-52814-0},
      url={https://www.springer.com/us/book/9783030528140},
      year={2021},
      publisher={Springer International Publishing}
}