Combinatorial Matrix Theory and Generalized Inverses of Matrices

Cover......Page 1
Combinatorial Matrix Theory and Generalized Inverses of Matrices......Page 3
Foreword......Page 5
Preface......Page 6
Acknowledgement......Page 7
Contents......Page 9
Editors......Page 11
Contributors......Page 12
1 Introduction......Page 15
2 Cartesian Product of Oriented Graphs with Hypercubes......Page 16
3 Structured Inverse Eigenvalue Problem (SIEP)......Page 20
4 Orientation of Hypercubes......Page 22
5 The Skew Energy of the Oriented Hypercube Qdphi......Page 24
References......Page 26
1 Introduction......Page 27
2 Singular Values......Page 28
3 q-Analog of EndSn(V(B(n)))......Page 31
4 Nonbinary Analog of EndSn(V(B(n)))......Page 36
References......Page 44
1 Introduction......Page 46
1.1 Immanants of Matrices......Page 47
2 The q-Analogue of T\'s Laplacian......Page 48
3 Laplacian of Connected r-Regular Graphs......Page 51
References......Page 53
1 Introduction......Page 54
2 Characterization......Page 56
3 Further Results......Page 60
3.1 Degree of a Vertex in Matrix Product......Page 61
3.2 Regular Graphs G for Which A(G)A(G) Is Graphical......Page 65
References......Page 68
1 Introduction......Page 69
2 Determinant of the Laplacian Matrix of a Weighted Directed Graph......Page 71
References......Page 74
1 Introduction......Page 75
2 Skew-Symmetric Distributions: Notions, Notation, Estimation......Page 76
3 Copulas from Skewed Distributions......Page 81
References......Page 83
1 Introduction......Page 85
2 Preliminaries......Page 87
3 When Is  PX; W1+y BLUE Under M2?......Page 89
References......Page 93
1 Introduction......Page 96
2 Homoscedastic Models......Page 98
3 EO Models......Page 100
4 Mixed Models......Page 104
5 An Application......Page 106
References......Page 110
1 Preliminaries......Page 112
2 Entries of Orthogonal Projectors......Page 114
3 Changes of Entries of Orthogonal Projectors......Page 121
4 Linear Regression Model......Page 122
References......Page 129
1 Introduction and Preliminaries......Page 130
2 Moore-Penrose Inverse of Rank-One Perturbed Operators......Page 132
3 Formulae for (A+V1GV2*)†......Page 135
4.1 Perturbation Bounds for Nonnegativity of Moore-Penrose Inverses......Page 139
4.2 Rank-One Modified Operator and Star Partial Order......Page 140
References......Page 142
1 Introduction......Page 143
2 Notation, Definitions, and Preliminaries......Page 145
3 Main Results......Page 146
References......Page 151
1 Introduction......Page 152
2 Some Estimates for Approximation Numbers......Page 154
3 Closed Range Operators and Approximation Numbers......Page 165
References......Page 167
1 Introduction......Page 168
1.2 Moore-Penrose Inverse and Its Condition Number......Page 169
2 Main Results......Page 172
3 Concluding Remark......Page 176
References......Page 177
1 Introduction......Page 179
2 Products of Idempotent Matrices......Page 180
3 Sums of Idempotent Matrices......Page 181
References......Page 182
1 Introduction......Page 184
3 Main Results......Page 185
4 Illustration......Page 187
References......Page 188
1 Introduction......Page 189
2 Preliminaries......Page 190
3 Invertible Matrices over an Incline......Page 191
4 Generalized Inverses of Matrices over an Incline......Page 195
References......Page 198
1 Preliminaries......Page 200
1.1 Matrices and Generalized Inverses......Page 201
1.2 Partial Order on a Set......Page 202
2 Partial Order on Matrices......Page 204
2.1 Column and Row Space-Dependent Relations......Page 205
2.2 Partial Ordering via Space Decomposition......Page 210
3 Drazin\'s Star Partial Order......Page 212
3.1 Lattice Properties of Star Order......Page 216
3.2 Left- and Right-Star Orders......Page 221
4.1 Minus Partial Order......Page 222
4.2 Minus Partial Order on the Class of Inner and Outer Inverses......Page 226
References......Page 230
An Illustrated Introduction to Some Old Magic Squares from India......Page 232
1 Introduction......Page 233
2 Daivajna Varahamihira (505-587 AD) and his Brhat Samhita [39]: Magic Perfume......Page 234
3 Khajuraho and Dudhai (Jhansi district)......Page 238
4 Thakkura Pheru (fl. 1291-1323)......Page 241
5 Robert Shortrede (1800-1868), Andrew Hollingworth Frost (1819-1907), and Two Gwalior Magic Matrices......Page 242
6 Srinivasa Aiyangar Ramanujan (1887-1920)......Page 245
References......Page 253
1.1 Objective of the Workshop......Page 258
1.3 Participation......Page 259
2.1 Societies and National Agencies......Page 260
2.2 Support from Manipal University......Page 261
4 International Workshop, 02-07 January 2012......Page 262
5 International Conference......Page 265
6 Conference Schedule and Activities......Page 267
7 Selected Photographs from the Events......Page 275
9 Message from Joint Secretary, CSIR......Page 278
10 Message from Director General, CSO......Page 279
Index......Page 280