## Matrix AnalysisA good part of matrix theory is functional analytic in spirit. This statement can be turned around. There are many problems in operator theory, where most of the complexities and subtleties are present in the finite-dimensional case. My purpose in writing this book is to present a systematic treatment of methods that are useful in the study of such problems. This book is intended for use as a text for upper division and gradu ate courses. Courses based on parts of the material have been given by me at the Indian Statistical Institute and at the University of Toronto (in collaboration with Chandler Davis). The book should also be useful as a reference for research workers in linear algebra, operator theory, mathe matical physics and numerical analysis. A possible subtitle of this book could be Matrix Inequalities. A reader who works through the book should expect to become proficient in the art of deriving such inequalities. Other authors have compared this art to that of cutting diamonds. One first has to acquire hard tools and then learn how to use them delicately. The reader is expected to be very thoroughly familiar with basic lin ear algebra. The standard texts Finite-Dimensional Vector Spaces by P.R. |

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### Contents

II | 1 |

III | 3 |

IV | 9 |

V | 12 |

VI | 16 |

VII | 20 |

VIII | 26 |

IX | 28 |

XL | 173 |

XLI | 181 |

XLII | 184 |

XLIII | 190 |

XLIV | 194 |

XLV | 195 |

XLVI | 203 |

XLVII | 211 |

X | 36 |

XI | 40 |

XII | 48 |

XIII | 50 |

XIV | 54 |

XV | 57 |

XVI | 62 |

XVII | 65 |

XVIII | 68 |

XIX | 73 |

XX | 75 |

XXI | 78 |

XXII | 84 |

XXIII | 91 |

XXIV | 98 |

XXV | 101 |

XXVI | 107 |

XXVII | 109 |

XXVIII | 112 |

XXIX | 117 |

XXX | 123 |

XXXI | 131 |

XXXII | 147 |

XXXIII | 149 |

XXXIV | 152 |

XXXV | 153 |

XXXVI | 155 |

XXXVII | 159 |

XXXVIII | 165 |

XXXIX | 168 |

### Common terms and phrases

Analysis arctan argument Bhatia called Chapter commutes complex numbers concave continuous convex function coordinates Corollary defined denote derivative diagonal entries diagonal matrix differentiable disk doubly stochastic matrix eigenvalues eigenvectors equivalent example Exercise exists finite function f(t generalisation given GL(n Hence Hermitian matrices Hermitian operator Hilbert space inner product interval invertible isotone Lemma Let A,B Let f Lidskii's Linear Algebra Linear Algebra Appl linear operator majorisation Math monotone function n x n nonnegative normal matrices notation Note obtained operator convex operator monotone operator monotone function operator norm orthogonal orthonormal basis permutation permutation matrices perturbation bounds polar decomposition polynomial positive matrices positive operators proof of Theorem Proposition proves the theorem Q-norm representation roots Section Show singular values skew-Hermitian subset subspaces symmetric gauge function Theory unitarily invariant norm unitary matrix upper triangular vector space Weyl's wui norm