| Preface | p. xv |
| Historical Introduction: Issai Schur and the Early Development of the Schur Complement | p. 1 |
| Introduction and mise-en-scene | p. 1 |
| The Schur complement: the name and the notation | p. 2 |
| Some implicit manifestations in the 1800s | p. 3 |
| The lemma and the Schur determinant formula | p. 4 |
| Issai Schur (1875-1941) | p. 6 |
| Schur's contributions in mathematics | p. 9 |
| Publication under J. Schur | p. 9 |
| Boltz 1923, Lohan 1933, Aitken 1937 and the Banchiewicz inversion formula 1937 | p. 10 |
| Frazer, Duncan & Collar 1938, Aitken 1939, and Duncan 1944 | p. 12 |
| The Aitken block-diagonalization formula 1939 and the Guttman rank additivity formula 1946 | p. 14 |
| Emilie Virginia Haynsworth (1916-1985) and the Haynsworth inertia additivity formula | p. 15 |
| Basic Properties of the Schur Complement | p. 17 |
| Notation | p. 17 |
| Gaussian elimination and the Schur complement | p. 17 |
| The quotient formula | p. 21 |
| Inertia of Hermitian matrices | p. 27 |
| Positive semidefinite matrices | p. 34 |
| Hadamard products and the Schur complement | p. 37 |
| The generalized Schur complement | p. 41 |
| Eigenvalue and Singular Value Inequalities of Schur Complements | p. 47 |
| Introduction | p. 47 |
| The interlacing properties | p. 49 |
| Extremal characterizations | p. 53 |
| Eigenvalues of the Schur complement of a product | p. 55 |
| Eigenvalues of the Schur complement of a sum | p. 64 |
| The Hermitian case | p. 69 |
| Singular values of the Schur complement of a product | p. 76 |
| Block Matrix Techniques | p. 83 |
| Introduction | p. 83 |
| Embedding approach | p. 85 |
| A matrix inequality and its applications | p. 92 |
| A technique by means of 2 x 2 block matrices | p. 99 |
| Liebian functions | p. 104 |
| Positive linear maps | p. 108 |
| Closure Properties | p. 111 |
| Introduction | p. 111 |
| Basic theory | p. 111 |
| Particular classes | p. 114 |
| Singular principal minors | p. 132 |
| Authors' historical notes | p. 136 |
| Schur Complements and Matrix Inequalities: Operator-Theoretic Approach | p. 137 |
| Introduction | p. 137 |
| Schur complement and orthoprojection | p. 140 |
| Properties of the map A [map to] [M]A | p. 148 |
| Schur complement and parallel sum | p. 152 |
| Application to the infimum problem | p. 157 |
| Schur Complements in Statistics and Probability | p. 163 |
| Basic results on Schur complements | p. 163 |
| Some matrix inequalities in statistics and probability | p. 171 |
| Correlation | p. 182 |
| The general linear model and multiple linear regression | p. 191 |
| Experimental design and analysis of variance | p. 213 |
| Broyden's matrix problem and mark-scaling algorithm | p. 221 |
| Schur Complements and Applications in Numerical Analysis | p. 227 |
| Introduction | p. 227 |
| Formal orthogonality | p. 228 |
| Pade application | p. 230 |
| Continued fractions | p. 232 |
| Extrapolation algorithms | p. 233 |
| The bordering method | p. 239 |
| Projections | p. 240 |
| Preconditioners | p. 248 |
| Domain decomposition methods | p. 250 |
| Triangular recursion schemes | p. 252 |
| Linear control | p. 257 |
| Bibliography | p. 259 |
| Notation | p. 289 |
| Index | p. 291 |
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