| Chapter 1 The field of values 1 1.0 Introduction 1 1.1 Definitions 5 1.2 Basic properties of the field of values 8 1.3 Convexity 17 1.4 Axiomatization 28 1.5 Location of the field of values 30 1.6 Geometry 48 1.7 Products of matrices 65 1.8 Generalizations of the field of values 77 Chapter 2 Stable matrices and inertia 89 2.0 Motivation 89 2.1 Definitions and elementary observations 91 2.2 Lyapunov''''''''s theorem 95 2.3 The Routh-Hurwitz conditions 101 2.4 Generalizations of Lyapunov''''''''s theorem 102 2.5 M-matrices, P-matrices, and related topics 112 Chapter 3 Singular value inequalities 134 3.0 Introduction and historical remarks 134 3.1 The singular value decomposition 144 3.2 Weak majorization and doubly substochastic matrices 163 3.3 Basic inequalities for singular values and eigenvalues 170 3.4 Sums of singular values: the Ky Fan k-norms 195 3.5 Singular values and unitarily invariant norms 203 3.6 Sufficiency of Weyl''''''''s product inequalities |
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