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Matrix Calculus and Kronecker Product

Willi-Hans Steeb and Yorick Hardy
World Scientific
Publication Date: 
Number of Pages: 
[Reviewed by
Michael Berg
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It’s a trite truism that linear algebra is a non-negotiable item in any mathematician’s toolkit. Its computational riches also provide plenty of opportunity for beginners to get their hands dirty — really dirty, if it’s done right — with dozens of projects centered on the marvelous behavior of matrices. And many of us get hooked: whether it be in the context of the representation theory of finite groups or that of the shenanigans the Frobenius automorphism pulls in étale cohomology, matrices and determinants take central stage. Throughout the spectrum of mathematical education and activity, from upper division undergraduate explorations through research level adventures, matrix calculus is a sine qua non.

Thus, the book under review, Matrix Calculus and Kronecker Product, by Steeb and Hardy of the University of Johannesburg, is poised to be of use in a significant manner. It introduces the reader to an extremely interesting and (I believe) underestimated player in linear and multilinear algebra. The Kronecker product of matrices is an extremely accessible (and cool) object and finds application is such diverse areas as quantum mechanics (cf. §§ 3.1–3.11 of the book: we encounter spin matrices, Pauli’s group, Fermi systems, and the Ising model, to list only a proper subset), signal processing (§ 3.13), Clebsch–Gordan series (§ 3.14), and Fast Fourier Transforms and image compression (§ 3.17). Furthermore, to show that it’s not only applied mathematics and physics that provide scope for this object, Steeb and Hardy’s second chapter, introducing the Kronecker product and accordingly titled, fits this construction into the framework of group representations (§§ 2.13, 2.14).

Lest the connection with analysis (well, functional analysis, of course) be neglected, the fourth and final chapter of the book under review, “Tensor product,” develops the theme of tensor products of Hilbert spaces, quickly heads in the direction of statistical mechanics and reaches a climactic conclusion with sections on interpreting Quantum Mechanics and universal enveloping algebras.

Naturally this is the second appearance of tensor products in the book: on p. 83, the opening page of Chapter 2, we read: “Sometimes the Kronecker product is also called direct product [rarely] or tensor product [Aye, there’s the rub … ]” The Kronecker product of two matrices, A and B, is obtained by forming a new matrix A⊗B constructed by replacing the each entry, aij of A by the matrix aijB. If we think of A as the matrix of an operator on a vector space V1 with respect to a basis {ek} and B as the matrix of an operator on V2 with respect to a basis {fl}, then A⊗B is the matrix of the corresponding operator on the vector space V1⊗V2 with respect to the basis {ek⊗fl} (correctly ordered).

Finally, regarding the first Chapter of the book, “Matrix Calculus,” well, it’s a smorgasbord of linear algebra viewed as the inner life of matrices. (Very different from what we’d find in, say, P. R. Halmos’ classic Finite Dimensional Vector Spaces: to be sure there’s a difference of opinion regarding how one should teach the subject to beginners. I’d like to suggest a compromise position: start ‘em off with Halmos, possibly doing some picking and choosing, and then head for Steeb and Hardy.). In Chapter 1 we find, e.g., eigenvalues, Cayley-Hamilton, Fourier and Hadamard matrices, Gram-Schmidt, and, as icing on the cake, some Lie theory.

Matrix Calculus and Kronecker Product is filled with examples and sports a decent set of exercise sets. The exposition is clear, if a little on the terse side, and the Preface provides an excellent roadmap for what’s to follow. All in all Steeb and Hardy have produced an excellent text for a second (or perhaps third) course in linear algebra. It should serve well indeed in all pedagogical contexts, be it a regular class-room course, a tutorial, or an REU. This is a valuable book (and already a success: it’s in its second edition) and, given the choice of applications in Chapters 3 and 4, even physics students get to play.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.

  • Matrix Calculus
  • Kronecker Product
  • Applications
  • Tensor Product
  • Computer Algebra Implementations