Quadratic form

About: Quadratic form is a research topic. Over the lifetime, 3192 publications have been published within this topic receiving 62231 citations.

Introduction to Matrix Analysis

Abstract: Foreword Preface to the Second Edition Preface 1. Maximization, Minimization, and Motivation 2. Vectors and Matrices 3. Diagonalization and Canonical Forms for Symmetric Matrices 4. Reduction of General Symmetric Matrices to Diagonal Form 5. Constrained Maxima 6. Functions of Matrices 7. Variational Description of Characteristic Roots 8. Inequalities 9. Dynamic Programming 10. Matrices and Differential Equations 11. Explicit Solutions and Canonical Forms 12. Symmetric Function, Kronecker Products and Circulants 13. Stability Theory 14. Markoff Matrices and Probability Theory 15. Stochastic Matrices 16. Positive Matrices, Perron's Theorem, and Mathematical Economics 17. Control Processes 18. Invariant Imbedding 19. Numerical Inversion of the Laplace Transform and Tychonov Regularization Appendix A. Linear Equations and Rank Appendix B. The Quadratic Form of Selberg Appendix C. A Method of Hermite Appendix D. Moments and Quadratic Forms Index.

Computing the distribution of quadratic forms in normal variables

Abstract: In this paper exact and approximate methods are given for computing the distribution of quadratic forms in normal variables. In statistical applications the interest centres in general, for a quadratic form Q and a given value x, around the probability P{Q > x}. Methods of computation have previously been given e.g. by Box (1954), Gurland (1955) and by Grad & Solomon (1955). None of these methods is very easily applicable except, when it can be used, the finite series of Box. Furthermore, all the methods are valid only for quadratic forms in central variables. Situations occur where quadratic forms in non-central variables must be considered as well. Let x = (x1, ..., xx)' be a column random vector which follows a multidimensional normal law with mean vector 0 and covariance matrix E. Let s = (,t, . . ., ,,7)' be a constant vector, and consider the quadratic form Q = (x + ,)' A(x + ,u). If E is non-singular, one can by means of a non-singular linear transformation (Scheff6 (1959), p. 418) express Q in the form rn 2 Q =E ArXhr; (1 r=1

An Application of the Shephard Duality Theorem: A Generalized Leontief Production Function

Abstract: The paper indicates how the Shephard duality theorem may be utilized in order to obtain a system of derived demand equations which are linear in the technological parameters, thus facilitating econometric estimation. This theorem states that technology may be equivalently represented by either a production function or a cost function, and a proof of the theorem is given. The chosen functional form is a quadratic form in the square roots of input prices and is a generalization of the Leontief cost function. The generalization has the property that it can attain any set of partial elasticities of substitution using a minimal number of parameters.

Introduction to quadratic forms

Abstract: Prerequisites ad Notation Part One: Arithmetic Theory of Fields I Valuated Fields Valuations Archimedean Valuations Non-Archimedean valuations Prolongation of a complete valuation to a finite extension Prolongation of any valuation to a finite separable extension Discrete valuations II Dedekind Theory of Ideals Dedekind axioms for S Ideal theory Extension fields III Fields of Number Theory Rational global fields Local fields Global fields Part Two: Abstract Theory of Quadratic Forms VI Quadratic Forms and the Orthogonal Group Forms, matrices and spaces Quadratic spaces Special subgroups of On(V) V The Algebras of Quadratic Forms Tensor products Wedderburn's theorem on central simple algebras Extending the field of scalars The clifford algebra The spinor norm Special subgroups of On(V) Quaternion algebras The Hasse algebra VI The Equivalence of Quadratic Forms Complete archimedean fields Finite fields Local fields Global notation Squares and norms in global fields Quadratic forms over global fields VII Hilbert's Reciprocity Law Proof of the reciprocity law Existence of forms with prescribed local behavior The quadratic reciprocity law Part Four: Arithmetic Theory of Quadratic Forms over Rings VIII Quadratic Forms over Dedekind Domains Abstract lattices Lattices in quadratic spaces IX Integral Theory of Quadratic Forms over Local Fields Generalities Classification of lattices over non-dyadic fields Classification of Lattices over dyadic fields Effective determination of the invariants Special subgroups of On(V) X Integral Theory of Quadratic Forms over Global Fields Elementary properties of the orthogonal group over arithmetic fields The genus and the spinor genus Finiteness of class number The class and the spinor genus in the indefinite case The indecomposable splitting of a definite lattice Definite unimodular lattices over the rational integers Bibliography Index Bibliography Index

Quadratic and Hermitian Forms

Abstract: 1. Basic Concepts.- 1. Bilinear Forms and Quadratic Forms.- 2. Matrix Notation.- 3. Regular Spaces and Orthogonal Decomposition.- 4. Isotropy and Hyperbolic Spaces.- 5. Witt's Theorem.- 6. Appendix: Symmetric Bilinear Forms and Quadratic Forms over Rings.- 2. Quadratic Forms over Fields.- 1. Grothendieck and Witt Rings.- 2. Invariants.- 3. Examples I (Finite Fields).- 4. Examples II (Ordered Fields).- 5. Ground Field Extension and Transfer.- 6. The Torsion of the Witt Group.- 7. Orderings, Pfister's Local Global Principle, and Prime Ideals of the Witt Ring.- 8. Applications of the Method of Transfer.- 9. Description of the Witt Ring by Generators and Relations.- 10. Multiplicative Forms.- 11. Quaternion Algebras.- 12. The Hasse Invariant and the Witt Invariant.- 13. The Hasse Algebra.- 14. Classification Theorems.- 15. Examples III. Ci-fields.- 16. The u-invariant.- 3. Quadratic Forms over Formally Real Fields.- 1. Formally Real and Ordered Fields.- 2. Real Closed Fields.- 3. Hilbert's 17th Problem and the Real Nullstellensatz.- 4. Extension of Signatures.- 5. The Space of Orderings of a Field.- 6. The Total Signature.- 7. A Local Global Principle for Weak Isotropy.- Appendix: Places, Valuations, and Valuation Rings.- 4. Generic Methods and Pfister Forms.- 1. Chain-p-equivalence of Pfister Forms.- 2. Pfister's Theorem on the Representation of Positive Functions as Sums of Squares.- 3. Casseis' and Pfister's Representation Theorems.- 4. Applications: Fields of Prescribed Level. Characterization of Pfister Forms.- 5. The Function Field of a Quadratic Form and the Main Theorem of Arason and Pfister.- 6. Generic Zeros and Generic Splitting.- 7. Knebusch's Filtration of the Witt Ring.- 5. Rational Quadratic Forms.- 1. Symmetric Bilinear Forms and Quadratic Forms on Finite Abelian Groups.- 2. Gaussian Sums for Quadratic Forms on Finite Abelian Groups.- 3. The Witt Group of1.- 4. The Witt Group of 2.- 5. Gauss' First Proof of the Quadratic Reciprocity Law.- 6. Quadratic Forms over the p-adic Numbers.- 7. Hilbert's Reciprocity Law and the Hasse-Minkowski Theorem.- 8. Calculation of Gaussian Sums.- 6. Symmetric Bilinear Forms over Dedekind Rings and Global Fields.- 1. Symmetric Bilinear Forms over Dedekind Rings.- 2. Symmetric Bilinear Forms over Discrete Valuation Rings.- 3. Symmetric Bilinear Forms over Polynomial Rings and Rational Function Fields.- 4. Symmetric Bilinear Forms over p-adic Fields.- 5. The Hilbert Reciprocity Theorem.- 6. The Hasse-Minkowski Theorem.- 7. Hecke's Theorem on the Different.- 8. The Residue Theorem.- 7. Foundations of the Theory of Hermitian Forms.- 1. Basic Definitions.- 2. Hermitian Categories.- 3. Quadratic Forms.- 4. Transfer and Reduction.- 5. Hermitian Abelian Categories.- 6. Hermitian Forms over Skew Fields.- 7. Hyperbolic Forms and the Unitary Group.- 8. Alternating Forms and the Symplectic Group.- 9. Witt's Theorem.- 10. The Krull-Schmidt Theorem.- 11. Examples and Applications.- 8. Simple Algebras and Involutions.- 1. Simple Rings and Modules.- 2. Tensor Products.- 3. Central Simple Algebras. The Brauer Group.- 4. Simple Algebras.- 5. Central Simple Algebras under Field Extensions. Reduced Norms and Traces.- 6. Examples.- 7. Involutions on Simple Algebras. The Classification Problem.- 8. Existence of Involutions.- 9. The Corestriction. Existence of Involutions of the Second Kind.- 10. An Extension Theorem for Involutions.- 11. Quaternion Algebras.- 12. Cyclic Algebras.- 13. The Canonical Involution on the Group Algebra.- 9. Clifford Algebras.- 1. Graded Algebras.- 2. Clifford Algebras.- 3. The Spinor Norm.- 4. Quadratic Forms over Fields in Characteristic 2.- 10. Hermitian Forms over Global Fields.- 1. Hermitian Forms over Commutative Fields and Quaternion Algebras.- 2. Simple Algebras and Involutions over Local and Global Fields.- 3. Skew Hermitian Forms over Quaternion Fields.- 4. Skew Hermitian Forms over Global Quaternion Fields..- 5. The Strong Approximation Theorem.- 6. Hermitian Forms for Unitary Involutions. Statement of Results.- 7. Proof of the Weak Local Global Principle.- 8. Conclusion of the Proof.