Series: Econometric Exercises
Hardback (ISBN: 0521822890)
Paperback (ISBN: 0521537460)
Lecturers can request inspection copies of this title.
Courses: matrix algebra, foundations of statistics and econometrics
Levels: 2ND YEAR UNDERGRADUATES AND ABOVE
Matrix Algebra is the first volume of the Econometric Exercises Series. It contains exercises relating to course material in matrix algebra that students are expected to know while enrolled in an (advanced) undergraduate or a postgraduate course in econometrics or statistics. The book contains a comprehensive collection of exercises, all with full answers. But the book is not just a collection of exercises; in fact, it is a textbook, though one that is organized in a completely different manner than the usual textbook. The volume can be used either as a self-contained course in matrix algebra or as a supplementary text.
■ Only text that is devoted to solved examples in matrix algebra
■ Nearly 400 pages of solved problems
■ Can be used either as a self-contained course in matrix algebra or as a supplementary text
Contents
1. Vectors: 1.1 Real vectors; 1.2 Complex vectors; 2. Matrices: 2.1 Real matrices; 2.2 Complex matrices; 3. Vector spaces: 3.1 Complex and real vector spaces; 3.2 Inner-product space; 3.3 Hilbert space; 4. Rank, inverse, and determinant: 4.1 Rank; 4.2 Inverse; 4.3 Determinant; 5. Partitioned matrices: 5.1 Basic results and multiplication relations; 5.2 Inverses; 5.3 Determinants; 5.4 Rank (in)equalities; 5.5 The sweep operator; 6. Systems of equations: 6.1 Elementary matrices; 6.2 Echelon matrices; 6.3 Gaussian elimination; 6.4 Homogeneous equations; 6.5 Nonhomogeneous equations; 7. Eigenvalues, eigenvectors, and factorizations: 7.1 Eigenvalues and eigenvectors; 7.2 Symmetric matrices; 7.3 Some results for triangular matrices; 7.4 Schur’s decomposition theorem and its consequences; 7.5 Jordan’s decomposition theorem; 7.6 Jordan chains and generalized eigenvectors; 8. Positive (semi)definite and idempotent matrices: 8.1 Positive (semi)definite matrices; 8.2 Partitioning and positive (semi)definite matrices; 8.3 Idempotent matrices; 9. Matrix functions: 9.1 Simple functions; 9.2 Jordan representation; 9.3 Matrix-polynomial representation; 10. Kronecker product, vec-operator, and Moore-Penrose inverse: 10.1 The Kronecker product; 10.2 The vec-operator; 10.3 The Moore-Penrose inverse; 10.4 Linear vector and matrix equations; 10.5 The generalized inverse; 11. Patterned matrices, commutation and duplication matrix: 11.1 The commutation matrix; 11.2 The symmetrizer matrix; 11.3 The vec-operator and the duplication matrix; 11.4 Linear structures; 12. Matrix inequalities: 12.1 Cauchy-Schwarz type inequalities; 12.2 Positive (semi)definite matrix inequalities; 12.3 Inequalities derived from the Schur complement; 12.4 Inequalities concerning eigenvalues; 13. Matrix calculus: 13.1 Basic properties of differentials; 13.2 Scalar functions; 13.3 Vector functions; 13.4 Matrix functions; 13.5 The inverse; 13.6 Exponential and logarithm; 13.7 The determinant; 13.8 Jacobians; 13.9 Sensitivity analysis in regression models; 13.10 The Hessian matrix; 13.11 Least squares and best linear unbiased estimation; 13.12 Maximum likelihood estimation; 13.13 Inequalities and equalities.
Reviews
'These authors have achieved the remarkable feat of writing a textbook of matrix algebra cunningly concealed as a structured sequence of exercises and worked answers. The book should prove popular with students intent on teaching themselves and with instructors who wish to set challenging and educative exercises. Recommended unequivocally to all parties.' Dr Stephen Pollock, Queen Mary College
'Useful as a text or reference, it is clearly written and very thorough. Besides basic topics, excellent treatment of matrix inequalities, vectorization, and matrix calculus. It belongs on every econometricians’s bookshelf.' Professor Peter Schmidt, Michigan State University
Series: Mathematical Sciences Research Institute Publications
Hardback (ISBN-10: 0521848628)
available from October 2005
During the past few decades, the gradual merger of Discrete Geometry and the newer discipline of Computational Geometry has provided enormous impetus to mathematicians and computer scientists interested in geometric problems. This volume, which contains 32 papers on a broad range of topics of current interest in the field, is an outgrowth of that synergism. It includes surveys and research articles exploring geometric arrangements, polytopes, packing, covering, discrete convexity, geometric algorithms and their complexity, and the combinatorial complexity of geometric objects, particularly in low dimension. There are points of contact with many applied areas such as mathematical programming, visibility problems, kinetic data structures, and biochemistry, as well as with algebraic topology, geometric probability, real algebraic geometry, and combinatorics.
■ Over 30 surveys and research papers by current leaders in the field
■ It contains several articles on new research areas, such as Geometric Graph Theory and the Theory of Core Sets and their applications
Contents
1. Geometric approximation via core sets Pankaj K. Agarwal, Sariel Har-Peled and Kasturi Varadarajan; 2. Applications of graph and hypergraph theory in geometry Imre Barany; 3. Convex geometry of orbits Alexander Barvinok and Grigoriy Blekherman; 4. The Hadwiger transversal theorem for pseudolines Saugata Basu, Jacob E. Goodman, Andreas Holmsen and Richard Pollack; 5. Betti number bounds, applications, and algorithms Saugata Basu, Richard Pollack and Marie-Francoise Roy; 6. Shelling and the h-vector of the (extra-)ordinary polytope Margaret M. Bayer; 7. On the number of mutually touching cylinders Andras Bezdek; 8. Edge-antipodal 3-polytopes Karoly Bezdek, Tibor Bisztriczky and Karoly Boroczky; 9. A conformal energy for simplicial surfaces Alexander Bobenko; 10. On the size of higher-dimensional triangulations Peter Brass; 11. The carpenter's ruler folding problem Gruia Calinescu and Adrian Dumitrescu; 12. A survey of folding and unfolding in computational geometry Erik D. Demaine and Joseph O'Rourke; 13. On the rank of a tropical matrix Mike Develin, Francisco Santos and Bernd Sturmfels; 14. The geometry of biomolecular solvation Herbert Edelsbrunner and Patrice Koehl; 15. Inequalities for zonotopes Richard G. Ehrenborg; 16. Quasiconvex programming David Eppstein; 17. De Concini-Procesi wonderful arrangement models - a discrete geometer's point of view Eva Maria Feichtner; 18. Thinnest covering of a circle by eight, nine, or ten congruent circles Gabor Fejes Toth; 19. On the complexity of visibility problems with moving viewpoints Peter Gritzmann and Thorsten Theobald; 20. Cylindrical partitions of convex bodies Aladar Heppes and Wlodzimierz Kuperberg; 21. Tropical halfspaces Michael Joswig; 22. Two proofs for Sylvester's problem using an allowable sequence of permutations Hagit Last; 23. A comparison of five implementations of 3d Delaunay tessellation Yuanxin Liu and Jack Snoeyink; 24. Bernstein's basis and real root isolation Bernard Mourrain, Fabrice Rouillier and Marie-Francoise Roy; 25. On some extremal problems in combinatorial geometry Niranjan Nilakantan; 26. A long non-crossing path among disjoint segments in the plane Janos Pach and Rom Pinchasi; 27. On a generalization of Schoenhardt's polyhedron Joerg Rambau; 28. On Hadwiger numbers of direct products of convex bodies Istvan Talata; 29. Binary space partitions - recent developments Csaba D. Toth; 30. Erdos-Szekeres theorem: upper bounds and related results Geza Toth and Pavel Valtr; 31. On the pair-crossing number Pavel Valtr; 32. Geometric random walks: a survey Santosh Vempala.
Contributors
Pankaj K. Agarwal, Sariel Har-Peled, Kasturi Varadarajan, Imre Barany, Alexander Barvinok, Grigoriy Blekherman, Saugata Basu, Jacob E. Goodman, Andreas Holmsen, Richard Pollack, Saugata Basu, Richard Pollack, Marie-Francoise Roy, Margaret M. Bayer, Andras Bezdek, Karoly Bezdek, Tibor Bisztriczky, Karoly Boroczky, Alexander Bobenko, Peter Brass, Gruia Calinescu, Adrian Dumitrescu, Erik D. Demaine, Joseph O'Rourke, Mike Develin, Francisco Santos, Bernd Sturmfels, Herbert Edelsbrunner, Patrice Koehl, Richard G. Ehrenborg, David Eppstein, Eva Maria Feichtner, Gabor Fejes Toth, Peter Gritzmann, Thorsten Theobald, Aladar Heppes, Wlodzimierz Kuperberg, Michael Joswig, Hagit Last, Yuanxin Liu, Jack Snoeyink, Bernard Mourrain, Fabrice Rouillier, Niranjan Nilakantan, Janos Pach, Rom Pinchasi, Joerg Rambau, Istvan Talata, Csaba D. Toth, Geza Toth, Pavel Valtr, Santosh Vempala
203 p. 6 x 9 2005
Series: (CLM) Chicago Lectures in Mathematics
Cloth 0-226-53983-0 Fall 2005
Paper 0-226-53984-9 Fall 2005
The theorems of Berkeley mathematician Marina Ratner have guided key advances in the understanding of dynamical systems. Unipotent flows are well-behaved dynamical systems, and Ratner has shown that the closure of every orbit for such a flow is of a simple algebraic or geometric form. In Ratner's Theorems on Unipotent Flows, Dave Witte Morris provides both an elementary introduction to these theorems and an account of the proof of Ratner's measure classification theorem.
A collection of lecture notes aimed at graduate students, the first four chapters of Ratner's Theorems on Unipotent Flows can be read independently. The first chapter, intended for a fairly general audience, provides an introduction with examples that illustrate the theorems, some of their applications, and the main ideas involved in the proof. In the following chapters, Morris introduces entropy, ergodic theory, and the theory of algebraic groups. The book concludes with a proof of the measure-theoretic version of Ratner's Theorem. With new material that has never before been published in book form, Ratner's Theorems on Unipotent Flows helps bring these important theorems to a broader mathematical readership.
Cloth | 2005 | ISBN: 0-691-11946-5
624 pp. | 6 x 9 | 289 line illus. 5 tables.
Pure and applied mathematicians, physicists, scientists, and engineers use matrices and operators and their eigenvalues in quantum mechanics, fluid mechanics, structural analysis, acoustics, ecology, numerical analysis, and many other areas. However, in some applications the usual analysis based on eigenvalues fails. For example, eigenvalues are often ineffective for analyzing dynamical systems such as fluid flow, Markov chains, ecological models, and matrix iterations. That's where this book comes in.
This is the authoritative work on nonnormal matrices and operators, written by the authorities who made them famous. Each of the sixty sections is written as a self-contained essay. Each document is a lavishly illustrated introductory survey of its topic, complete with beautiful numerical experiments and all the right references. The breadth of included topics and the numerous applications that provide links between fields will make this an essential reference in mathematics and related sciences.
Lloyd N. Trefethen is Professor of Numerical Analysis and Head of the Numerical Analysis Group at the University of Oxford. Mark Embree is Assistant Professor of Computational and Applied Mathematics at Rice University.
ISBN: 0-415-29802-4
Pub Date: 28 JUL 2003
Type: Hardback Book
Extent: 448 pages(Dimensions X mm)
Part of an extensive two volume set, this work contains some of the most important papers by the renowned geometer A.D. Alexandrov, including an examination of general propositions on an intrinsic metric, angles and curvature, the existence of a convex polyhedron with prescribed metric, curves on convex surfaces, and the role of specific curvature.
Consisting of an original core of classic material, originally published in Russian in 1948, as well as additional supplementary information, this detailed work will appeal to advanced students and researchers in mathematics and geometry.
Contents:
Basic Concepts and Results. General Propositions about the Intrinsic Metric. Characteristic Properties of the Intrinsic Metric. Angle. Curvature. Existence of a Convex Polyhedron with a Given Metric. Existence of a Closed Convex Surface with a Given Metric. Other Existence Theorems. Curves on Convex Surfaces. Area. The Role of the Specific Curvature. Generalization. Appendix. Basic Facts of Convex Bodies.
Series Information:
Classics of Soviet Mathematics
Full Contributors:
Kutateladze; S.S. Novosibirsk State University , Russia,