A Matrix Handbook for StatisticiansISBN: 9780471748694
559 pages
November 2007

This timely book, A Matrix Handbook for Statisticians, provides a comprehensive, encyclopedic treatment of matrices as they relate to both statistical concepts and methodologies. Written by an experienced authority on matrices and statistical theory, this handbook is organized by topic rather than mathematical developments and includes numerous references to both the theory behind the methods and the applications of the methods. A uniform approach is applied to each chapter, which contains four parts: a definition followed by a list of results; a short list of references to related topics in the book; one or more references to proofs; and references to applications. The use of extensive crossreferencing to topics within the book and external referencing to proofs allows for definitions to be located easily as well as interrelationships among subject areas to be recognized.
A Matrix Handbook for Statisticians addresses the need for matrix theory topics to be presented together in one book and features a collection of topics not found elsewhere under one cover. These topics include:

Complex matrices

A wide range of special matrices and their properties

Special products and operators, such as the Kronecker product

Partitioned and patterned matrices

Matrix analysis and approximation

Matrix optimization

Majorization

Random vectors and matrices

Inequalities, such as probabilistic inequalities
Additional topics, such as rank, eigenvalues, determinants, norms, generalized inverses, linear and quadratic equations, differentiation, and Jacobians, are also included. The book assumes a fundamental knowledge of vectors and matrices, maintains a reasonable level of abstraction when appropriate, and provides a comprehensive compendium of linear algebra results with use or potential use in statistics. A Matrix Handbook for Statisticians is an essential, oneofakind book for graduatelevel courses in advanced statistical studies including linear and nonlinear models, multivariate analysis, and statistical computing. It also serves as an excellent selfstudy guide for statistical researchers.
1. Notation.
1.1 General Definitions.
1.2 Some Continuous Univariate Distributions.
1.3 Glossary of Notation.
2. Vectors, Vector Spaces, and Convexity.
2.1 Vector Spaces.
2.1.1 Definitions.
2.1.2 Quadratic Subspaces.
2.1.3 Sums and Intersections of Subspaces.
2.1.4 Span and Basis.
2.1.5 Isomorphism.
2.2 Inner Products.
2.2.1 Definition and Properties.
2.2.2 Functionals.
2.2.3 Orthogonality.
2.2.4 Column and Null Spaces.
2.3 Projections.
2.3.1 General Projections.
2.3.2 Orthogonal Projections.
2.4 Metric Spaces.
2.5 Convex Sets and Functions.
2.6 Coordinate Geometry.
2.6.1 Hyperplanes and Lines.
2.6.2 Quadratics.
2.6.3 Miscellaneous Results.
3. Rank.
3.1 Some General Properties.
3.2 Matrix Products.
3.3 Matrix Cancellation Rules.
3.4 Matrix Sums.
3.5 Matrix Differences.
3.6 Partitioned Matrices.
3.7 Maximal and Minimal Ranks.
3.8 Matrix Index.
4. Matrix Functions: Inverse, Transpose, Trace, Determinant, and Norm.
4.1 Inverse.
4.2 Transpose.
4.3 Trace.
4.4 Determinants.
4.4.1 Introduction.
4.4.2 Adjoint Matrix.
4.4.3 Compound Matrix.
4.4.4 Expansion of a Determinant.
4.5 Permanents.
4.6 Norms.
4.6.1 Vector Norms.
4.6.2 Matrix Norms.
4.6.3 Unitarily Invariant Norms.
4.6.4 M,NInvariant Norms.
4.6.5 Computational Accuracy.
5. Complex, Hermitian, and Related Matrices.
5.1 Complex Matrices.
5.1.1 Some General Results.
5.1.2 Determinants.
5.2 Hermitian Matrices.
5.3 SkewHermitian Matrices.
5.4 Complex Symmetric Matrices.
5.5 Real SkewSymmetric Matrices.
5.6 Normal Matrices.
5.7 Quaternions.
6. Eigenvalues, Eigenvectors, and Singular Values.
6.1 Introduction and Definitions.
6.1.1 Characteristic Polynomial.
6.1.2 Eigenvalues.
6.1.3 Singular Values.
6.1.4 Functions of a Matrix.
6.1.5 Eigenvectors.
6.1.6 Hermitian Matrices.
6.1.7 Computational Methods.
6.1.8 Generalized Eigenvalues.
6.1.9 Matrix Products 103.
6.2 Variational Characteristics for Hermitian Matrices.
6.3 Separation Theorems.
6.4 Inequalities for Matrix Sums.
6.5 Inequalities for Matrix Differences.
6.6 Inequalities for Matrix Products.
6.7 Antieigenvalues and Antieigenvectors.
7. Generalized Inverses.
7.1 Definitions.
7.2 Weak Inverses.
7.2.1 General Properties.
7.2.2 Products.
7.2.3 Sums and Differences.
7.2.4 Real Symmetric Matrices.
7.2.5 Decomposition Methods.
7.3 Other Inverses.
7.3.1 Reflexive (g12) Inverse.
7.3.2 Minimum Norm (g14) Inverse.
7.3.3 Minimum Norm Reflexive (g124) Inverse.
7.3.4 Least Squares (g13) Inverse.
7.3.5 Least Squares Reflexive (g123) Inverse.
7.4 MoorePenrose (g1234) Inverse.
7.4.1 General Properties.
7.4.2 Sums.
7.4.3 Products.
7.5 Group Inverse.
7.6 Some General Properties of Inverses.
8. Some Special Matrices.
8.1 Orthogonal and Unitary Matrices.
8.2 Permutation Matrices.
8.3 Circulant, Toeplitz, and Related Matrices.
8.3.1 Regular Circulant.
8.3.2 Symmetric Regular Circulant.
8.3.3 Symmetric Circulant.
8.3.4 Toeplitz Matrix.
8.3.5 Persymmetric Matrix.
8.3.6 CrossSymmetric (Centrosymmetric) Matrix.
8.3.7 Block Circulant.
8.3.8 Hankel Matrix.
8.4 Diagonally Dominant Matrices.
8.5 Hadamard Matrices.
8.6 Idempotent Matrices.
8.6.1 General Properties.
8.6.2 Sums of Idempotent Matrices and Extensions.
8.6.3 Products of Idempotent Matrices.
8.7 Tripotent Matrices.
8.8 Irreducible Matrices.
8.9 Triangular Matrices.
8.10 Hessenberg Matrices.
8.11 Tridiagonal Matrices.
8.12 Vandermonde and Fourier Matrices.
8.12.1 Vandermonde Matrix.
8.12.2 Fourier Matrix.
8.13 ZeroOne (0,1) Matrices.
8.14 Some Miscellaneous Matrices and Arrays.
8.14.1 Krylov Matrix.
8.14.2 Nilpotent and Unipotent Matrices.
8.14.3 Payoff Matrix.
8.14.4 Stable and Positive Stable Matrices.
8.14.5 PMatrix.
8.14.6 Z and MMatrices.
8.14.7 ThreeDimensional Arrays.
9. NonNegative Vectors and Matrices.
9.1 Introduction.
9.1.1 Scaling.
9.1.2 Modulus of a Matrix.
9.2 Spectral Radius.
9.2.1 General Properties.
9.2.2 Dominant Eigenvalue.
9.3 Canonical Form of a Nonnegative Matrix.
9.4 Irreducible Matrices.
9.4.1 Irreducible Nonnegative Matrix.
9.4.2 Periodicity.
9.4.3 Nonnegative and Nonpositive OffDiagonal Elements.
9.4.4 Perron Matrix.
9.4.5 Decomposable Matrix.
9.5 Leslie Matrix.
9.6 Stochastic Matrices.
9.6.1 Basic Properties.
9.6.2 Finite Homogeneous Markov Chain.
9.6.3 Countably Infinite Stochastic Matrix.
9.6.4 Infinite Irreducible Stochastic Matrix.
9.7 Doubly Stochastic Matrices.
10. Positive Definite and Nonnegative Definite Matrices.
10.1 Introduction.
10.2 Nonnegative Definite Matrices.
10.2.1 Some General Properties.
10.2.2 Gram Matrix.
10.2.3 Doubly Nonnegative Matrix.
10.3 Positive Definite Matrices.
10.4 Pairs of Matrices.
10.4.1 NonNegative or Positive Definite Difference.
10.4.2 One or More NonNegative Definite Matrices.
11. Special Products and Operators.
11.1 Kronecker Product.
11.1.1 Two Matrices.
11.1.2 More Than Two Matrices.
11.2 Vec Operator.
11.3 VecPermutation (Commutation) Matrix.
11.4 Generalized VecPermutation Matrix.
11.5 Vech Operator.
11.5.1 Symmetric Matrix.
11.5.2 Lower Triangular Matrix.
11.6 Star Operator.
11.7 Hadamard Product.
11.8 RaoKhatri Product.
12. Inequalities.
12.1 CauchySchwarz inequalities.
12.1.1 Real Vector Inequalities and Extensions.
12.1.2 Complex Vector Inequalities.
12.1.3 Real Matrix Inequalities.
12.1.4 Complex Matrix Inequalities.
12.2 H?older?s Inequality and Extensions.
12.3 Minkowski?s Inequality and Extensions.
12.4 Weighted Means.
12.5 Quasilinearization (Representation Theorems).
12.6 Some Geometrical Properties.
12.7 Miscellaneous Inequalities.
12.7.1 Determinants.
12.7.2 Trace.
12.7.3 Quadratics.
12.7.4 Sums and Products.
12.8 Some Identities.
13. Linear Equations.
13.1 Unknown vector.
13.1.1 Consistency.
13.1.2 Solutions.
13.1.3 Homogeneous Equations.
13.1.4 Restricted Equations.
13.2 Unknown Matrix.
13.2.1 Consistency.
13.2.2 Some Special Cases.
14. Partitioned Matrices.
14.1 Schur Complement.
14.2 Inverses.
14.3 Determinants.
14.4 Positive and NonNegative Definite matrices.
14.5 Eigenvalues.
14.6 Generalized Inverses.
14.6.1 Weak Inverses.
14.6.2 MoorePenrose Inverses.
14.7 Miscellaneous partitions.
15. Patterned Matrices.
15.1 Inverses.
15.2 Determinants.
15.3 Perturbations.
15.4 Matrices With Repeated Elements and Blocks.
15.5 Generalized Inverses.
15.5.1 Weak Inverses.
15.5.2 MoorePenrose Inverses.
16. Factorization of Matrices.
16.1 Similarity Reductions.
16.2 Reduction by Elementary Transformations.
16.2.1 Types of Transformation.
16.2.2 Equivalence Relation.
16.2.3 Echelon Form.
16.2.4 Hermite Form.
16.3 Singular Value Decomposition (SVD).
16.4 Triangular Factorizations.
16.5 OrthogonalTriangular Reductions.
16.6 Further Diagonal or Tridiagonal Reductions.
16.7 Congruence.
16.8 Simultaneous Reductions.
16.9 Polar Decomposition.
16.10 Miscellaneous Factorizations.
17. Differentiation and Finite Differences.
17.1 Introduction.
17.2 Scalar Differentiation.
17.2.1 Differentiation with Respect to t.
17.2.2 Differentiation With Respect to a Vector Element.
17.2.3 Differentiation With Respect to a Matrix Element.
17.3 Vector Differentiation: Scalar Function.
17.3.1 Basic Results.
17.3.2 x=vec X.
17.3.3 Function of a Function.
17.4 Vector Differentiation: Vector Function.
17.5 Matrix Differentiation: Scalar Function.
17.5.1 General Results.
17.5.2 f = trace.
17.5.3 f = determinant.
17.5.4 f = yrs.
17.5.5 f = eigenvalue.
17.6 Transformation Rules.
17.7 Matrix Differentiation: Matrix Function.
17.8 Matrix Differentials.
17.9 Perturbation Using Differentials.
17.10 Matrix Linear Differential Equations.
17.11 Second Order Derivatives.
17.12 Vector Difference Equations.
18. Jacobians.
18.1 Introduction.
18.2 Method of Differentials.
18.3 Further Techniques.
18.3.1 Chain Rule.
18.3.2 Exterior (Wedge) Product of Differentials.
18.3.3 Induced Functional Equations.
18.3.4 Jacobians Involving Transposes.
18.3.5 Patterned Matrices and LStructures.
18.4 Vector Transformations.
18.5 Jacobians for Complex Vectors and Matrices.
18.6 Matrices with Functionally Independent Elements.
18.7 Symmetric and Hermitian Matrices.
18.8 SkewSymmetric and SkewHermitian Matrices.
18.9 Triangular Matrices.
18.9.1 Linear Transformations.
18.9.2 Nonlinear Transformations of X.
18.9.3 Decompositions With One matrix Skew Symmetric.
18.9.4 Symmetric Y.
18.9.5 Positive Definite Y.
18.9.6 Hermitian Positive Definite Y.
18.9.7 Skew Symmetric Y.
18.9.8 LU Decomposition.
18.10 Decompositions Involving Diagonal Matrices.
18.10.1 Square Matrices.
18.10.2 One Triangular Matrix.
18.10.3 Symmetric and Skew Symmetric Matrices.
18.11 Positive?Definite Matrices.
18.12 Caley Transformation.
18.13 Diagonalizable Matrices.
18.14 Pairs of Matrices.
19. Matrix Limits, Sequences and Series.
19.1 Limits.
19.2 Sequences.
19.3 Asymptotically Equivalent Sequences.
19.4 Series.
19.5 Matrix Functions.
19.6 Matrix Exponentials.
20. Random Vectors.
20.1 Notation.
20.2 Variances and Covariances.
20.3 Correlations.
20.3.1 Population Correlations.
20.3.2 Sample Correlations.
20.4 Quadratics.
20.5 Multivariate Normal Distribution.
20.5.1 Definition and Properties.
20.5.2 Quadratics in
20.5.3 Quadratics and Chisquared.
20.5.4
20.5.5
20.6 Complex Random Vectors.
20.7 Regression Models.
20.7.1 V is the Identity Matrix.
20.7.2 V is Positive Definite.
20.7.3 V is Nonnegative Definite.
20.8 Other Multivariate Distributions.
20.8.1 Multivariate tDistribution.
20.8.2 Elliptical and Spherical Distributions.
20.8.3 Dirichlet Distributions.
21. Random Matrices.
21.1 Introduction.
21.2 Generalized Quadratic Forms.
21.2.1 General Results.
21.2.2 Wishart Distribution.
21.3 Random Samples.
21.3.1 One Sample.
21.3.2 Two Samples.
21.4 Multivariate Linear Model.
21.4.1 Least Squares Estimation.
21.4.2 Statistical Inference.
21.4.3 Two Extensions.
21.5 Dimension Reduction Techniques.
21.5.1 Principal Component Analysis (PCA).
21.5.2 Discriminant Coordinates.
21.5.3 Canonical Correlations and Variates.
21.5.4 Latent Variable Methods.
21.5.5 Classical (Metric) Scaling.
21.6 Procrustes Analysis (Matching Configurations).
21.7 Some Specific Random Matrices.
21.8 Allocation Problems.
21.9 Matrix Variate Distributions.
21.10 Matrix Ensembles.
22. Inequalities for Probabilities and Random Variables.
22.1 General Probabilities.
22.2 BonferroniType Inequalities.
22.3 DistributionFree Probability Inequalities.
22.3.1 ChebyshevType Inequalities.
22.3.2 KolmogorovType Inequalities.
22.3.3 Quadratics and Inequalities.
22.4 Data Inequalities.
22.5 Inequalities for Expectations.
22.6 Multivariate Inequalities.
22.6.1 Convex Subsets.
22.6.2 Multivariate
22.6.3 Inequalities For Other Distributions.
23. Majorization.
23.1 General Properties.
23.2 Schur Convexity.
23.3 Probabilities and Random variables.
24. Optimization and Matrix Approximation.
24.1 Stationary Values.
24.2 Using Convex and Concave Functions.
24.3 Two General Methods.
24.3.1 Maximum Likelihood.
24.3.2 Least Squares.
24.4 Optimizing a Function of a Matrix.
24.4.1 Trace.
24.4.2 Norm.
24.4.3 Quadratics.
24.5 Optimal Designs.
References.
Index.
George A. F. Seber, PhD, is Emeritus Professor in the Department of Statistics at The University of Auckland in New Zealand. A Fellow of the New Zealand Royal Society, he is the author or coauthor of several books, including Nonlinear Regression, Multivariate Observations, Adaptive Sampling, Chance Encounters, and Linear Regression Analysis, Second Edition, all published by Wiley. Dr. Seber's research interests have included statistical ecology, genetics, epidemiology, and adaptive sampling.
"This book maintains its uniqueness among the competition through its extensive referencing to proofs and comprehensive coverage of topics not found in any other one book." (International Statistical Review, Dec 2008)
"This is an authoritative and comprehensive reference that will be useful to researchers who need to use the results of matrix analysis in their work. It would also be a useful addition to the reference collection of any mathematical library." (MAA Review, March 2008)
"This is an authoritative and comprehensive reference that will be useful to researchers who need to use the results of matrix analysis in their work. It would also be a useful addition to the reference collection of any mathematical library." (MAA Review, March 2008)
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