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**Matrix inequalities.**
*(English)*
Zbl 1018.15016

Lecture Notes in Mathematics. 1790. Berlin: Springer. vii, 116 p. EUR 22.95/net; sFr. 39.50; £16.00; $ 29.80 (2002).

Matrix inequalities can be considered as matrix analysis from the quantitative point of view. Anyone working in this field is aware of both the charming beauty of the theory and the intricacy of its proofs. This situation is excellently reflected by this monograph.

It consists of five chapters. The first one establishes inequalities in the Löwner partial order. Chapter 2 introduces several types of majorization and deals with their applications to eigenvalues of the Hadamard product of positive semidefinite matrices (thus extending Oppenheim’s inequality). Chapters 3 and 4 constitute the main body of the book; they are devoted to inequalities for singular values resp. norms. The last chapter is an overview on the famous solution of the van der Waerden conjecture concerning the minimum permanent of doubly stochastic matrices.

The author succeeds in giving a comprehensive account of, more or less, recent trends and developments in the field of matrix inequalities, and focusing on useful techniques and ingenious ideas (using tools from classical analysis, algebra and combinatorics), thus presenting the current state of the art of research. The reader encounters the affirmative solutions of not less than eight conjectures as well as a number of theorems unifying previous results. Except a few preliminary results, there is no overlap with the standard monographs by R. A. Horn and C. R. Johnson [Topics in matrix analysis (1991; Zbl 0729.15001)] and R. Bhatia [Matrix analysis, Graduate Texts in Mathematics, 169 (1996; Zbl 0863.15001)].

The presentation is very clear and a pleasure to read. At the end of each section, one finds references to the literature that serve as a useful guide to the origin and development of the topics dealt with. The book is almost self-contained; it requires basic knowledge in linear algebra, real and complex analysis and, moreover, some familiarity with the mentioned books by Horn and Johnson resp. Bhatia. It can be recommended to graduate students, advanced undergraduate students and researchers in matrix analysis, as well.

It consists of five chapters. The first one establishes inequalities in the Löwner partial order. Chapter 2 introduces several types of majorization and deals with their applications to eigenvalues of the Hadamard product of positive semidefinite matrices (thus extending Oppenheim’s inequality). Chapters 3 and 4 constitute the main body of the book; they are devoted to inequalities for singular values resp. norms. The last chapter is an overview on the famous solution of the van der Waerden conjecture concerning the minimum permanent of doubly stochastic matrices.

The author succeeds in giving a comprehensive account of, more or less, recent trends and developments in the field of matrix inequalities, and focusing on useful techniques and ingenious ideas (using tools from classical analysis, algebra and combinatorics), thus presenting the current state of the art of research. The reader encounters the affirmative solutions of not less than eight conjectures as well as a number of theorems unifying previous results. Except a few preliminary results, there is no overlap with the standard monographs by R. A. Horn and C. R. Johnson [Topics in matrix analysis (1991; Zbl 0729.15001)] and R. Bhatia [Matrix analysis, Graduate Texts in Mathematics, 169 (1996; Zbl 0863.15001)].

The presentation is very clear and a pleasure to read. At the end of each section, one finds references to the literature that serve as a useful guide to the origin and development of the topics dealt with. The book is almost self-contained; it requires basic knowledge in linear algebra, real and complex analysis and, moreover, some familiarity with the mentioned books by Horn and Johnson resp. Bhatia. It can be recommended to graduate students, advanced undergraduate students and researchers in matrix analysis, as well.

Reviewer: Arnold Richard Kräuter (Leoben)

### MSC:

15A45 | Miscellaneous inequalities involving matrices |

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

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

15A18 | Eigenvalues, singular values, and eigenvectors |

15A42 | Inequalities involving eigenvalues and eigenvectors |

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

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

26D15 | Inequalities for sums, series and integrals |

15B51 | Stochastic matrices |