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**Matrix mathematics. Theory, facts, and formulas.
2nd expanded ed.**
*(English)*
Zbl 1183.15001

Princeton, NJ: Princeton University Press (ISBN 978-0-691-14039-1/pbk; 978-0-691-13287-7/hbk). xxxix, 1139 p. (2009).

As I have mentioned in my review Zbl 1075.15001 for the first version of this book (2005), the author has succeeded to gather and present in a very nice way an impressive collection of matrix facts that appear through the scientific, mathematical and engineering literature into a reference source. It is a necessary book for anyone using matrices in his research. It contains the majority of the results that concerns the matrix theory usually in terms of facts with all the necessary bibliography.

The second edition is a major expansion of the first edition. This new edition includes extra material on scalar identities and inequalities; trigonometric and hyperbolic identities; graphs (developed within the framework of relations, and partially ordered sets); matrices related to graphs; partitioned matrices; matrix representation of finite groups; matrix pencils and Kronecker canonical forms; zeros of multivariable transfer functions; vector and matrix norms. Due to the extra material, the number of cited works has been increased at 87%.

The second edition is a major expansion of the first edition. This new edition includes extra material on scalar identities and inequalities; trigonometric and hyperbolic identities; graphs (developed within the framework of relations, and partially ordered sets); matrices related to graphs; partitioned matrices; matrix representation of finite groups; matrix pencils and Kronecker canonical forms; zeros of multivariable transfer functions; vector and matrix norms. Due to the extra material, the number of cited works has been increased at 87%.

Reviewer: Nicholas Karampetakis (Thessaloniki)

### MSC:

15-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to linear algebra |

93-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to systems and control theory |

15A04 | Linear transformations, semilinear transformations |

15A06 | Linear equations (linear algebraic aspects) |

15A09 | Theory of matrix inversion and generalized inverses |

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

15A16 | Matrix exponential and similar functions of matrices |

93B05 | Controllability |

93B07 | Observability |

93C05 | Linear systems in control theory |

93D05 | Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory |

15A18 | Eigenvalues, singular values, and eigenvectors |

15A21 | Canonical forms, reductions, classification |

15A24 | Matrix equations and identities |

15B48 | Positive matrices and their generalizations; cones of matrices |

15A54 | Matrices over function rings in one or more variables |

15B57 | Hermitian, skew-Hermitian, and related matrices |

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