Moment-Sequence Transforms
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Linear Algebra: Example Sheet 2 of 4
Michaelmas Term 2014 SJW Linear Algebra: Example Sheet 2 of 4 1. (Another proof of the row rank column rank equality.) Let A be an m × n matrix of (column) rank r. Show that r is the least integer for which A factorises as A = BC with B 2 Matm;r(F) and C 2 Matr;n(F). Using the fact that (BC)T = CT BT , deduce that the (column) rank of AT equals r. 2. Write down the three types of elementary matrices and find their inverses. Show that an n × n matrix A is invertible if and only if it can be written as a product of elementary matrices. Use this method to find the inverse of 0 1 −1 0 1 @ 0 0 1 A : 0 3 −1 3. Let A and B be n × n matrices over a field F . Show that the 2n × 2n matrix IB IB C = can be transformed into D = −A 0 0 AB by elementary row operations (which you should specify). By considering the determinants of C and D, obtain another proof that det AB = det A det B. 4. (i) Let V be a non-trivial real vector space of finite dimension. Show that there are no endomorphisms α; β of V with αβ − βα = idV . (ii) Let V be the space of infinitely differentiable functions R ! R. Find endomorphisms α; β of V which do satisfy αβ − βα = idV . 5. Find the eigenvalues and give bases for the eigenspaces of the following complex matrices: 0 1 1 0 1 0 1 1 −1 1 0 1 1 −1 1 @ 0 3 −2 A ; @ 0 3 −2 A ; @ −1 3 −1 A : 0 1 0 0 1 0 −1 1 1 The second and third matrices commute; find a basis with respect to which they are both diagonal. -
Matrix-Valued Moment Problems
Matrix-valued moment problems A Thesis Submitted to the Faculty of Drexel University by David P. Kimsey in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics June 2011 c Copyright 2011 David P. Kimsey. All Rights Reserved. ii Acknowledgments I wish to thank my advisor Professor Hugo J. Woerdeman for being a source of mathematical and professional inspiration. Woerdeman's encouragement and support have been extremely beneficial during my undergraduate and graduate career. Next, I would like to thank Professor Robert P. Boyer for providing a very important men- toring role which guided me toward a career in mathematics. I owe many thanks to L. Richard Duffy for providing encouragement and discussions on fundamental topics. Professor Dmitry Kaliuzhnyi-Verbovetskyi participated in many helpful discussions which helped shape my understanding of several mathematical topics of interest to me. In addition, Professor Anatolii Grinshpan participated in many useful discussions gave lots of encouragement. I wish to thank my Ph.D. defense committee members: Professors Boyer, Lawrence A. Fialkow, Grinshpan, Pawel Hitczenko, and Woerde- man for pointing out any gaps in my understanding as well as typographical mistakes in my thesis. Finally, I wish to acknowledge funding for my research assistantship which was provided by the National Science Foundation, via the grant DMS-0901628. iii Table of Contents Abstract ................................................................................ iv 1. INTRODUCTION ................................................................ -
Linear Algebra Review James Chuang
Linear algebra review James Chuang December 15, 2016 Contents 2.1 vector-vector products ............................................... 1 2.2 matrix-vector products ............................................... 2 2.3 matrix-matrix products ............................................... 4 3.2 the transpose .................................................... 5 3.3 symmetric matrices ................................................. 5 3.4 the trace ....................................................... 6 3.5 norms ........................................................ 6 3.6 linear independence and rank ............................................ 7 3.7 the inverse ...................................................... 7 3.8 orthogonal matrices ................................................. 8 3.9 range and nullspace of a matrix ........................................... 8 3.10 the determinant ................................................... 9 3.11 quadratic forms and positive semidefinite matrices ................................ 10 3.12 eigenvalues and eigenvectors ........................................... 11 3.13 eigenvalues and eigenvectors of symmetric matrices ............................... 12 4.1 the gradient ..................................................... 13 4.2 the Hessian ..................................................... 14 4.3 gradients and hessians of linear and quadratic functions ............................. 15 4.5 gradients of the determinant ............................................ 16 4.6 eigenvalues -
A New Algorithm to Obtain the Adjugate Matrix Using CUBLAS on GPU González, H.E., Carmona, L., J.J
ISSN: 2277-3754 ISO 9001:2008 Certified International Journal of Engineering and Innovative Technology (IJEIT) Volume 7, Issue 4, October 2017 A New Algorithm to Obtain the Adjugate Matrix using CUBLAS on GPU González, H.E., Carmona, L., J.J. Information Technology Department, ININ-ITTLA Information Technology Department, ININ proved to be the best option for most practical applications. Abstract— In this paper a parallel code for obtain the Adjugate The new transformation proposed here can be obtained from Matrix with real coefficients are used. We postulate a new linear it. Next, we briefly review this topic. transformation in matrix product form and we apply this linear transformation to an augmented matrix (A|I) by means of both a minimum and a complete pivoting strategies, then we obtain the II. LU-MATRICIAL DECOMPOSITION WITH GT. Adjugate matrix. Furthermore, if we apply this new linear NOTATION AND DEFINITIONS transformation and the above pivot strategy to a augmented The problem of solving a linear system of equations matrix (A|b), we obtain a Cramer’s solution of the linear system of Ax b is central to the field of matrix computation. There are equations. That new algorithm present an O n3 computational several ways to perform the elimination process necessary for complexity when n . We use subroutines of CUBLAS 2nd A, b R its matrix triangulation. We will focus on the Doolittle-Gauss rd and 3 levels in double precision and we obtain correct numeric elimination method: the algorithm of choice when A is square, results. dense, and un-structured. nxn Index Terms—Adjoint matrix, Adjugate matrix, Cramer’s Let us assume that AR is nonsingular and that we rule, CUBLAS, GPU. -
Field of U-Invariants of Adjoint Representation of the Group GL (N, K)
Field of U-invariants of adjoint representation of the group GL(n,K) K.A.Vyatkina A.N.Panov ∗ It is known that the field of invariants of an arbitrary unitriangular group is rational (see [1]). In this paper for adjoint representation of the group GL(n,K) we present the algebraically system of generators of the field of U-invariants. Note that the algorithm for calculating generators of the field of invariants of an arbitrary rational representation of an unipotent group was presented in [2, Chapter 1]. The invariants was constructed using induction on the length of Jordan-H¨older series. In our case the length is equal to n2; that is why it is difficult to apply the algorithm. −1 Let us consider the adjoint representation AdgA = gAg of the group GL(n,K), where K is a field of zero characteristic, on the algebra of matri- ces M = Mat(n,K). The adjoint representation determines the representation −1 ρg on K[M] (resp. K(M)) by formula ρgf(A)= f(g Ag). Let U be the subgroup of upper triangular matrices in GL(n,K) with units on the diagonal. A polynomial (rational function) f on M is called an U- invariant, if ρuf = f for every u ∈ U. The set of U-invariant rational functions K(M)U is a subfield of K(M). Let {xi,j} be a system of standard coordinate functions on M. Construct a X X∗ ∗ X X∗ X∗ X matrix = (xij). Let = (xij) be its adjugate matrix, · = · = det X · E. Denote by Jk the left lower corner minor of order k of the matrix X. -
Geometry of the Pfaff Lattices 3
GEOMETRY OF THE PFAFF LATTICES YUJI KODAMA∗ AND VIRGIL U. PIERCE∗∗ Abstract. The (semi-infinite) Pfaff lattice was introduced by Adler and van Moerbeke [2] to describe the partition functions for the random matrix models of GOE and GSE type. The partition functions of those matrix models are given by the Pfaffians of certain skew-symmetric matrices called the moment matrices, and they are the τ-functions of the Pfaff lattice. In this paper, we study a finite version of the Pfaff lattice equation as a Hamiltonian system. In particular, we prove the complete integrability in the sense of Arnold-Liouville, and using a moment map, we describe the real isospectral varieties of the Pfaff lattice. The image of the moment map is a convex polytope whose vertices are identified as the fixed points of the flow generated by the Pfaff lattice. Contents 1. Introduction 2 1.1. Lie algebra splitting related to SR-factorization 2 1.2. Hamiltonian structure 3 1.3. Outline of the paper 5 2. Integrability 6 2.1. The integrals Fr,k(L) 6 2.2. The angle variables conjugate to Fr,k(L) 10 2.3. Extended Jacobian 14 3. Matrix factorization and the τ-functions 16 3.1. Moment matrix and the τ-functions 17 3.2. Foliation of the phase space by Fr,k(L) 21 3.3. Examples from the matrix models 22 4. Real solutions 23 arXiv:0705.0510v1 [nlin.SI] 3 May 2007 4.1. Skew-orthogonal polynomials 23 4.2. Fixed points of the Pfaff flows 26 4.3. -
Formalizing the Matrix Inversion Based on the Adjugate Matrix in HOL4 Liming Li, Zhiping Shi, Yong Guan, Jie Zhang, Hongxing Wei
Formalizing the Matrix Inversion Based on the Adjugate Matrix in HOL4 Liming Li, Zhiping Shi, Yong Guan, Jie Zhang, Hongxing Wei To cite this version: Liming Li, Zhiping Shi, Yong Guan, Jie Zhang, Hongxing Wei. Formalizing the Matrix Inversion Based on the Adjugate Matrix in HOL4. 8th International Conference on Intelligent Information Processing (IIP), Oct 2014, Hangzhou, China. pp.178-186, 10.1007/978-3-662-44980-6_20. hal-01383331 HAL Id: hal-01383331 https://hal.inria.fr/hal-01383331 Submitted on 18 Oct 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Distributed under a Creative Commons Attribution| 4.0 International License Formalizing the Matrix Inversion Based on the Adjugate Matrix in HOL4 Liming LI1, Zhiping SHI1?, Yong GUAN1, Jie ZHANG2, and Hongxing WEI3 1 Beijing Key Laboratory of Electronic System Reliability Technology, Capital Normal University, Beijing 100048, China [email protected], [email protected] 2 College of Information Science & Technology, Beijing University of Chemical Technology, Beijing 100029, China 3 School of Mechanical Engineering and Automation, Beihang University, Beijing 100083, China Abstract. This paper presents the formalization of the matrix inversion based on the adjugate matrix in the HOL4 system. -
Pre-Publication Accepted Manuscript
Peter Forrester, Shi-Hao Li Classical discrete symplectic ensembles on the linear and exponential lattice: skew orthogonal polynomials and correlation functions Transactions of the American Mathematical Society DOI: 10.1090/tran/7957 Accepted Manuscript This is a preliminary PDF of the author-produced manuscript that has been peer-reviewed and accepted for publication. It has not been copyedited, proofread, or finalized by AMS Production staff. Once the accepted manuscript has been copyedited, proofread, and finalized by AMS Production staff, the article will be published in electronic form as a \Recently Published Article" before being placed in an issue. That electronically published article will become the Version of Record. This preliminary version is available to AMS members prior to publication of the Version of Record, and in limited cases it is also made accessible to everyone one year after the publication date of the Version of Record. The Version of Record is accessible to everyone five years after publication in an issue. CLASSICAL DISCRETE SYMPLECTIC ENSEMBLES ON THE LINEAR AND EXPONENTIAL LATTICE: SKEW ORTHOGONAL POLYNOMIALS AND CORRELATION FUNCTIONS PETER J. FORRESTER AND SHI-HAO LI Abstract. The eigenvalue probability density function for symplectic invariant random matrix ensembles can be generalised to discrete settings involving either a linear or exponential lattice. The corresponding correlation functions can be expressed in terms of certain discrete, and q, skew orthogonal polynomials respectively. We give a theory of both of these classes of polynomials, and the correlation kernels determining the correlation functions, in the cases that the weights for the corresponding discrete unitary ensembles are classical. -
Linearized Polynomials Over Finite Fields Revisited
Linearized polynomials over finite fields revisited∗ Baofeng Wu,† Zhuojun Liu‡ Abstract We give new characterizations of the algebra Ln(Fqn ) formed by all linearized polynomials over the finite field Fqn after briefly sur- veying some known ones. One isomorphism we construct is between ∨ L F n F F n n( q ) and the composition algebra qn ⊗Fq q . The other iso- morphism we construct is between Ln(Fqn ) and the so-called Dickson matrix algebra Dn(Fqn ). We also further study the relations between a linearized polynomial and its associated Dickson matrix, generalizing a well-known criterion of Dickson on linearized permutation polyno- mials. Adjugate polynomial of a linearized polynomial is then intro- duced, and connections between them are discussed. Both of the new characterizations can bring us more simple approaches to establish a special form of representations of linearized polynomials proposed re- cently by several authors. Structure of the subalgebra Ln(Fqm ) which are formed by all linearized polynomials over a subfield Fqm of Fqn where m|n are also described. Keywords Linearized polynomial; Composition algebra; Dickson arXiv:1211.5475v2 [math.RA] 1 Jan 2013 matrix algebra; Representation. ∗Partially supported by National Basic Research Program of China (2011CB302400). †Key Laboratory of Mathematics Mechanization, AMSS, Chinese Academy of Sciences, Beijing 100190, China. Email: [email protected] ‡Key Laboratory of Mathematics Mechanization, AMSS, Chinese Academy of Sciences, Beijing 100190, China. Email: [email protected] 1 1 Introduction n Let Fq and Fqn be the finite fields with q and q elements respectively, where q is a prime or a prime power. -
Arxiv:2012.03523V3 [Math.NT] 16 May 2021 Favne Cetfi Optn Eerh(SR Spr Ft of Part (CM4)
WRONSKIAN´ ALGEBRA AND BROADHURST–ROBERTS QUADRATIC RELATIONS YAJUN ZHOU To the memory of Dr. W (1986–2020) ABSTRACT. Throughalgebraicmanipulationson Wronskian´ matrices whose entries are reducible to Bessel moments, we present a new analytic proof of the quadratic relations conjectured by Broadhurst and Roberts, along with some generalizations. In the Wronskian´ framework, we reinterpret the de Rham intersection pairing through polynomial coefficients in Vanhove’s differential operators, and compute the Betti inter- section pairing via linear sum rules for on-shell and off-shell Feynman diagrams at threshold momenta. From the ideal generated by Broadhurst–Roberts quadratic relations, we derive new non-linear sum rules for on-shell Feynman diagrams, including an infinite family of determinant identities that are compatible with Deligne’s conjectures for critical values of motivic L-functions. CONTENTS 1. Introduction 1 2. W algebra 5 2.1. Wronskian´ matrices and Vanhove operators 5 2.2. Block diagonalization of on-shell Wronskian´ matrices 8 2.3. Sumrulesforon-shellandoff-shellBesselmoments 11 3. W⋆ algebra 20 3.1. Cofactors of Wronskians´ 21 3.2. Threshold behavior of Wronskian´ cofactors 24 3.3. Quadratic relations for off-shell Bessel moments 30 4. Broadhurst–Roberts quadratic relations 37 4.1. Quadratic relations for on-shell Bessel moments 37 4.2. Arithmetic applications to Feynman diagrams 43 4.3. Alternative representations of de Rham matrices 49 References 53 arXiv:2012.03523v3 [math.NT] 16 May 2021 Date: May 18, 2021. Keywords: Bessel moments, Feynman integrals, Wronskian´ matrices, Bernoulli numbers Subject Classification (AMS 2020): 11B68, 33C10, 34M35 (Primary) 81T18, 81T40 (Secondary) * This research was supported in part by the Applied Mathematics Program within the Department of Energy (DOE) Office of Advanced Scientific Computing Research (ASCR) as part of the Collaboratory on Mathematics for Mesoscopic Modeling of Materials (CM4). -
Lecture 5: Matrix Operations: Inverse
Lecture 5: Matrix Operations: Inverse • Inverse of a matrix • Computation of inverse using co-factor matrix • Properties of the inverse of a matrix • Inverse of special matrices • Unit Matrix • Diagonal Matrix • Orthogonal Matrix • Lower/Upper Triangular Matrices 1 Matrix Inverse • Inverse of a matrix can only be defined for square matrices. • Inverse of a square matrix exists only if the determinant of that matrix is non-zero. • Inverse matrix of 퐴 is noted as 퐴−1. • 퐴퐴−1 = 퐴−1퐴 = 퐼 • Example: 2 −1 0 1 • 퐴 = , 퐴−1 = , 1 0 −1 2 2 −1 0 1 0 1 2 −1 1 0 • 퐴퐴−1 = = 퐴−1퐴 = = 1 0 −1 2 −1 2 1 0 0 1 2 Inverse of a 3 x 3 matrix (using cofactor matrix) • Calculating the inverse of a 3 × 3 matrix is: • Compute the matrix of minors for A. • Compute the cofactor matrix by alternating + and – signs. • Compute the adjugate matrix by taking a transpose of cofactor matrix. • Divide all elements in the adjugate matrix by determinant of matrix 퐴. 1 퐴−1 = 푎푑푗(퐴) det(퐴) 3 Inverse of a 3 x 3 matrix (using cofactor matrix) 3 0 2 퐴 = 2 0 −2 0 1 1 0 −2 2 −2 2 0 1 1 0 1 0 1 2 2 2 0 2 3 2 3 0 Matrix of Minors = = −2 3 3 1 1 0 1 0 1 0 2 3 2 3 0 0 −10 0 0 −2 2 −2 2 0 2 2 2 1 −1 1 2 −2 2 Cofactor of A (퐂) = −2 3 3 .∗ −1 1 −1 = 2 3 −3 0 −10 0 1 −1 1 0 10 0 2 2 0 adj A = CT = −2 3 10 2 −3 0 2 2 0 0.2 0.2 0 1 1 A-1 = ∗ adj A = −2 3 10 = −0.2 0.3 1 |퐴| 10 2 −3 0 0.2 −0.3 0 4 Properties of Inverse of a Matrix • (A-1)-1 = A • (AB)-1 = B-1A-1 • (kA)-1 = k-1A-1 where k is a non-zero scalar. -
Representational Models: a Common Framework for Understanding Encoding, Pattern-Component, and Representational-Similarity Analysis
Representational models: A common framework for understanding encoding, pattern-component, and representational-similarity analysis Jörn Diedrichsen1 & Nikolaus Kriegeskorte2 1. Brain and Mind Institute, Western University, Canada 2. Cognitive and Brain Sciences Unit, Cambridge University, UK Abstract Representational models explain how activity patterns in populations of neurons (or, more generally, in multivariate brain activity measurements) relate to sensory stimuli, motor actions, or cognitive processes. In an experimental context, representational models can be defined as probabilistic hypotheses about what profiles of activations across conditions are likely to be observed. We describe three methods to test such models – encoding models, pattern component modeling (PCM), and representational similarity analysis (RSA). We show that these methods are closely related in that they all evaluate the statistical second moment of the activity profile distribution. Using simulated data from three different fMRI experiments, we compare the power of the approaches to adjudicate between competing representational models. PCM implements a likelihood-ratio test and therefore constitutes the most powerful test if its assumptions hold. However, the other two approaches – when conducted appropriately – can perform similarly. In encoding approaches, the linear model needs to be appropriately regularized, which imposes a prior on the activity profiles. Without such a prior, encoding approaches do not test well-defined representational models. In RSA, the unequal variances and dependencies of the distance measures can be taken into account using a multi-normal approximation to the sampling distribution of the distance measures, so as to enable optimal inference. Each of the three techniques renders different information explicit (e.g. single response tuning in encoding models and population representational dissimilarity in RSA) and has specific advantages in terms of computational demands, ease of use, and extensibility.