Contents

Preface xv

Chapter 0 Historical Introduction: Issai Schur and the Early Development of the Schur Complement 1

Simo Puntanen, University of Tampere, Tampere, Finland George P. H. Styan, McGill University, Montreal, Canada

0.0 Introduction and mise-en-scene 1

0.1 The Schur complement: the name and the notation 2

0.2 Some implicit manifestations in the 1800s 3

0.3 The lemma and the Schur determinant formula 4

0.4 Issai Schur (1875-1941) 6

0.5 Schur's contributions in mathematics 9

0.6 Publication under J. Schur 9

0.7 Boltz 1923, Lohan 1933, Aitken 1937 and the Banchiewicz inversion formula 1937 10

0.8 Frazer, Duncan & Collar 1938, Aitken 1939, and Duncan 1944 12

0.9 The Aitken block-diagonalization formula 1939 and the Guttman rank additivity formula 1946 14

0.10 Emilie Virginia Haynsworth (1916-1985) and the Haynsworth inertia additivity formula 15

Chapter 1 Basic Properties of the Schur Complement 17

Roger A. Horn, University of Utah, Salt Lake City, USA Fuzhen Zhang, Nova Southeastern University, Fort Lauderdale, USA and Shenyang Normal University, Shenyang, China

1.0 Notation 17

1.1 Gaussian elimination and the Schur complement 17

1.2 The quotient formula 21

1.3 Inertia of Hermitian matrices 27

1.4 Positive semidefinite matrices 34

1.5 Hadamard products and the Schur complement .37

Chapter 2 Eigenvalue and Singular Value Inequalities of Schur Complements 47

Jianzhou Liu, Xiangtang University, Xiangtang, China

2.0 Introduction 47

2.1 The interlacing properties 49

2.2 Extremal characterizations 53

2.3 Eigenvalues of the Schur complement of a product 55

2.4 Eigenvalues of the Schur complement of a sum 64

2.5 The Hermitian case 69

2.6 Singular values of the Schur complement of a product 76

Chapter 3 Block Matrix Techniques 83

Fuzhen Zhang, Nova Southeastern University, Fort Lauderdale, USA and Shenyang Normal University, Shenyang, China

3.0 Introduction 83

3.1 Embedding approach 85

3.2 A matrix inequality and its applications 92

3.3 A technique by means of 2 x 2 block matrices 99

3.4 Liebian functions 104

3.5 Positive linear maps 108

Chapter 4 Closure Properties 111

Charles R. Johnson, College of William and Mary, Williamsburg, USA Ronald L. Smith, University of Tennessee, Chattanooga, USA

4.0 Introduction Ill

4.1 Basic theory Ill

4.2 Particular classes 114

4.3 Singular principal minors 132

4.4 Authors' historical notes 136

Chapter 5 Schur Complements and Matrix Inequalities: Operator-Theoretic Approach 137

Tsuyoshi Ando, Hokkaido University, Sapporo, Japan

5.0 Introduction 137

5.1 Schur complement and orthoprojection 140

5.2 Properties of the map A ^ [M]A 148

5.3 Schur complement and parallel sum 152

Chapter 6 Schur Complements in Statistics and Probability 163

Simo Puntanen, University of Tampere, Tampere, Finland George P. H. Styan, McGill University, Montreal, Canada

6.0 Basic results on Schur complements 163

6.1 Some matrix inequalities in statistics and probability 171

6.2 Correlation 182

6.3 The general linear model and multiple linear regression 191

6.4 Experimental design and analysis of variance 213

6.5 Broyden's matrix problem and mark-scaling algorithm 221

Chapter 7 Schur Complements and Applications in Numerical Analysis 227

Claude Brezinski, Universite des Sciences et Technologies de Lille, France

7.0 Introduction 227

7.1 Formal orthogonality 228

7.2 Fade application 230

7.3 Continued fractions 232

7.4 Extrapolation algorithms 233

7.5 The bordering method 239

7.6 Frojections 240

7.7 Freconditioners 248

7.8 Domain decomposition methods 250

7.9 Triangular recursion schemes 252

7.10 Linear control 257

Bibliography 259

Notation 289