Topics in quaternion linear algebra.

*(English)*Zbl 1304.15004
Princeton Series in Applied Mathematics. Princeton, NJ: Princeton University Press (ISBN 978-0-691-16185-3/hbk; 978-1-400-85274-1/ebook). xii, 363 p. (2014).

This book is an advanced text covering the entire area of linear algebra and matrix analysis over the skew field of real quaternions. It is thus a book for specialists and assumes a knowledge of linear algebra over the real and complex numbers. Proofs that carry over unchanged are omitted. Though it deals with a specialized topic, the exposition is accessible to upper undergraduate students of mathematics and can thus also be used for supplying supplementary material for linear algebra courses.

The book consists essentially of two parts. The first (Chapters 2–7) introduces quaternions and covers the fundamentals of linear algebra over the quaternions, including matrix decompositions, numerical ranges, Jordan and Kronecker canonical forms, canonical forms under congruence, determinants and invariant subspaces. In the second part (Chapters 8–14), the exposition is more like a research monograph and deals with canonical forms of quaternion matrix pencils with symmetries or, what is the same, pairs of matrices with symmetries. Applications are given to systems of linear differential equations with symmetries and to matrix equations. For the benefit of students, the last chapter (Chapter 15) is an appendix containing summaries of material used in the book, namely Jordan and Kronecker canonical forms, and real as well as complex matrix pencils with symmetries. More than 200 exercises ranging from the routine to open-ended questions are provided.

This is the first book dedicated to a systematic exposition of quaternion linear algebra and represents the state of the art, though inevitably some topics are not covered, e.g., those pertaining to numerical analysis, and orthogonal, unitary and symplectic quaternion matrices are dealt with only briefly. As quaternions are becoming increasingly useful in applications of theoretical and applied mathematics, this book will be very much welcome by researchers and students alike.

The book consists essentially of two parts. The first (Chapters 2–7) introduces quaternions and covers the fundamentals of linear algebra over the quaternions, including matrix decompositions, numerical ranges, Jordan and Kronecker canonical forms, canonical forms under congruence, determinants and invariant subspaces. In the second part (Chapters 8–14), the exposition is more like a research monograph and deals with canonical forms of quaternion matrix pencils with symmetries or, what is the same, pairs of matrices with symmetries. Applications are given to systems of linear differential equations with symmetries and to matrix equations. For the benefit of students, the last chapter (Chapter 15) is an appendix containing summaries of material used in the book, namely Jordan and Kronecker canonical forms, and real as well as complex matrix pencils with symmetries. More than 200 exercises ranging from the routine to open-ended questions are provided.

This is the first book dedicated to a systematic exposition of quaternion linear algebra and represents the state of the art, though inevitably some topics are not covered, e.g., those pertaining to numerical analysis, and orthogonal, unitary and symplectic quaternion matrices are dealt with only briefly. As quaternions are becoming increasingly useful in applications of theoretical and applied mathematics, this book will be very much welcome by researchers and students alike.

Reviewer: Rabe von Randow (Bonn)

##### MSC:

15-02 | Research exposition (monographs, survey articles) pertaining to linear algebra |

15B33 | Matrices over special rings (quaternions, finite fields, etc.) |

12E15 | Skew fields, division rings |

11R52 | Quaternion and other division algebras: arithmetic, zeta functions |

16K99 | Division rings and semisimple Artin rings |

15A60 | Norms of matrices, numerical range, applications of functional analysis to matrix theory |

15A21 | Canonical forms, reductions, classification |

15A15 | Determinants, permanents, traces, other special matrix functions |

15A22 | Matrix pencils |

15A24 | Matrix equations and identities |