Albert W. Marshall. Ingram Olkin Barry. C. Arnold. Inequalities: Theory. of Majorization and Its Applications. Second Edition.

Transcription

1 Albert W Marshall Ingram Olkin Barry C Arnold Inequalities: Theory of Majorization and Its Applications Second Edition f) Springer

2 Contents I Theory of Majorization 1 Introduction 3 A Motivation and Basic Definitions 3 B Majorization as a Partial Ordering 18 C Order-Preserving Functions 19 D Various Generalizations of Majorization 21 2 Doubly Stochastic Matrices 29 A Doubly Stochastic Matrices and Permutation Matrices 29 B C D Characterization of Majorization Using Doubly Stochastic Matrices 32 Doubly Substochastic Matrices and Weak Majorization 36 Doubly Superstochastic Matrices and Weak Majorization 42 E Orderings on 45 F Proofs of Birkhoff's Theorem and Refinements 47 G Classes of Doubly Stochastic Matrices 52 xvii

3 xviii Contents More Examples of Doubly Stochastic and Doubly Substochastic Matrices 61 I Properties of Doubly Stochastic Matrices 67 J Diagonal Equivalence of Nonnegative Matrices 76 3 Schur-Convex Functions 79 A Characterization of Schur-Convex Functions B Compositions Involving Schur-Convex Functions C Some General Classes of Schur-Convex Functions D Examples I Sums of Convex Functions 101 E Examples II Products of Logarithmically Concave (Convex) Functions 105 F Examples III Elementary Symmetric Functions 114 G Muirhead's Theorem 120 Schur-Convex Functions on $ and Their Extension to Mn 132 I Miscellaneous Specific Examples 138 J Integral Transformations Preserving Schur- Convexity 145 K Physical Interpretations of Inequalities Equivalent Conditions for Majorization 155 A Characterization by Linear Transformations 155 B Characterization in Terms of Order-Preserving Functions 156 C A Geometric Characterization 162 D A Characterization Involving Top Wage Earners Preservation and Generation of Majorization 165 A Operations Preserving Majorization 165 B Generation of Majorization 185 C Maximal and Minimal Vectors Under Constraints 192 D Majorization in Integers 194 E Partitions 199 F Linear Transformations That Preserve Majorization Rearrangements and Majorization 203 A Majorizations from Additions of Vectors 204 B Majorizations from Functions of Vectors 210 C Weak Majorizations from Rearrangements 213 D L-Superadditive Functions Properties and Examples 217

4 Contents xix E Inequalities Without Majorization 225 P A Relative Arrangement Partial Order 228 II Mathematical Applications 7 Combinatorial Analysis 243 A Some Preliminaries on Graphs, Incidence Matrices, and Networks 243 B Conjugate Sequences 245 C The Theorem of Gale and Ryser 249 D Some Applications of the Gale-Ryser Theorem 254 E s-graphs and a Generalization of the Gale-Ryser Theorem 258 F Tournaments 260 G Edge Coloring in Graphs 265 Some Graph Theory Settings in Which Majorization Plays a Role Geometric Inequalities 269 A Inequalities for the Angles of a Triangle 271 B Inequalities for the Sides of a Triangle 276 C Inequalities for the Exradii and Altitudes 282 D Inequalities for the Sides, Exradii, and Medians E Isoperimetric-Type Inequalities for Plane Figures F Duality Between Triangle Inequalities and Inequalities Involving Positive Numbers 294 G Inequalities for Polygons and Simplexes Matrix Theory 297 A Notation and Preliminaries 298 B C Diagonal Elements and Eigenvalues of a ermitian Ma trix 300 Eigenvalues of a ermitian Matrix and Its Principal Submatrices 308 D Diagonal Elements and Singular Values 313 E Absolute Value of Eigenvalues and Singular Values 317 F Eigenvalues and Singular Values 324 G Eigenvalues and Singular Values of A, B, imda + B 329 Eigenvalues and Singular Values of A, B, and AB I Absolute Values of Eigenvalues and Row Sums

5 xx Contents J Schur or adamard Products of Matrices 352 K A Totally Positive Matrix and an M-Matrix 357 L Loewner Ordering and Majorization 360 M Nonnegative Matrix-Valued Functions 361 N Zeros of Polynomials Other Settings in Matrix Theory Where Majorization as Proved Useful Numerical Analysis 367 A Unitarily Invariant Norms and Symmetric Gauge Functions 367 B Matrices Closest to a Given Matrix 370 C Condition Numbers and Linear Equations 376 D Condition Numbers of Submatrices and Augmented Matrices 380 E Condition Numbers and Norms 380 III Stochastic Applications 11 Stochastic Majorizations 387 A Introduction 387 B Convex Functions and Exchangeable Random Variables 392 C Families of Distributions Parameterized to Preserve Symmetry and Convexity 397 D Some Consequences of the Stochastic Majorization E\(P{) E Parameterization to Preserve Schur-Convexity F Additional Stochastic Majorizations and Properties 420 G Weak Stochastic Majorizations 427 Additional Stochastic Weak Majorizations and Properties Stochastic Schur-Convexity Probabilistic, Statistical, and Other Applications 441 A Sampling from a Finite Population 442 B Majorization Using Jensen's Inequality 456 C Probabilities of Realizing at Least k of n Events 457 D Expected Values of Ordered Random Variables 461 E Eigenvalues of a Random Matrix 469 F Special Results for Bernoulli and Geometric Random Variables 474

6 Contents xxi G Weighted Sums of Symmetric Random Variables 476 Stochastic Ordering from Ordered Random Variables 481 I Another Stochastic Majorization Based on Stochas tic Ordering 487 J Peakedness of Distributions of Linear Combinations 490 K Tail Probabilities for Linear Combinations 494 L Schur-Concave Distribution Functions and Survival Functions 500 M Bivariate Probability Distributions with Fixed Marginals 505 N Combining Random Variables Concentration Inequalities for Multivariate Distributions 510 P Miscellaneous Cameo Appearances of Majorization 511 Q Some Other Settings in Which Majorization Plays a Role Additional Statistical Applications 527 A Unbiasedness of Tests and Monotonicity of Power Functions 528 B Linear Combinations of Observations 535 C Ranking and Selection 541 D Majorization in Reliability Theory 549 E Entropy 556 F Measuring Inequality and Diversity 559 G Schur-Convex Likelihood Functions 566 Probability Content of Geometric Regions for Schur-Concave Densities Optimal Experimental Design 568 J Comparison of Experiments 570 IV Generalizations 14 Orderings Extending Majorization 577 A Majorization with Weights 578 B Majorization Relative to d 585 C Semigroup and Group Majorization 587 D Partial Orderings Induced by Convex Cones 595 E Orderings Derived from Function Sets 598 F Other Relatives of Majorization 603

7 xxii Contents G Majorization with Respect to a Partial Order 605 Rearrangements and Majorizations for Functions Multivariate Majorization 611 A Some Basic Orders 611 B The Order-Preserving Functions 621 C Majorization for Matrices of Differing Dimensions 623 D Additional Extensions 628 E Probability Inequalities 630 V Complementary Topics 16 Convex Functions and Some Classical Inequalities 637 A Monotone Functions 637 B Convex Functions 641 C Jensen's Inequality 654 D Some Additional Fundamental Inequalities 657 E Matrix-Monotone and Matrix-Convex Functions 670 F Real-Valued Functions of Matrices Stochastic Ordering 693 A Some Basic Stochastic Orders 694 B Stochastic Orders from Convex Cones 700 C The Lorenz Order 712 D Lorenz Order: Applications and Related Results 734 E An Uncertainty Order Total Positivity 757 A Totally Positive Functions 757 B Polya Frequency Functions 762 C Polya Frequency Sequences 767 D Total Positivity of Matrices Matrix Factorizations, Compounds, Direct Products, and M-Matrices 769 A Eigenvalue Decompositions 769 B Singular Value Decomposition 771 C Square Roots and the Polar Decomposition 772 D A Duality Between Positive Semidefinite ermitian Matrices 774 E Simultaneous Reduction of Two ermitian Matrices 775

8 Contents xxiii F Compound Matrices 775 G Kronecker Product and Sum 780 M-Matrices Extremal Representations of Matrix Functions 783 A Eigenvalues of a ermitian Matrix 783 B Singular Values 789 C Other Extremal Representations 794 Biographies 797 References 813 Author Index 879 Subject Index 893