A Formal Proof of the Computation of Hermite Normal Form in a General Setting?
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User's Guide to Pari-GP
User's Guide to PARI / GP C. Batut, K. Belabas, D. Bernardi, H. Cohen, M. Olivier Laboratoire A2X, U.M.R. 9936 du C.N.R.S. Universit´eBordeaux I, 351 Cours de la Lib´eration 33405 TALENCE Cedex, FRANCE e-mail: [email protected] Home Page: http://www.parigp-home.de/ Primary ftp site: ftp://megrez.math.u-bordeaux.fr/pub/pari/ last updated 5 November 2000 for version 2.1.0 Copyright c 2000 The PARI Group Permission is granted to make and distribute verbatim copies of this manual provided the copyright notice and this permission notice are preserved on all copies. Permission is granted to copy and distribute modified versions, or translations, of this manual under the conditions for verbatim copying, provided also that the entire resulting derived work is distributed under the terms of a permission notice identical to this one. PARI/GP is Copyright c 2000 The PARI Group PARI/GP is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation. It is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY WHATSOEVER. Table of Contents Chapter 1: Overview of the PARI system . 5 1.1 Introduction . 5 1.2 The PARI types . 6 1.3 Operations and functions . 9 Chapter 2: Specific Use of the GP Calculator . 13 2.1 Defaults and output formats . 14 2.2 Simple metacommands . 20 2.3 Input formats for the PARI types . 23 2.4 GP operators . -
CONSTRUCTING INTEGER MATRICES with INTEGER EIGENVALUES CHRISTOPHER TOWSE,∗ Scripps College
Applied Probability Trust (25 March 2016) CONSTRUCTING INTEGER MATRICES WITH INTEGER EIGENVALUES CHRISTOPHER TOWSE,∗ Scripps College ERIC CAMPBELL,∗∗ Pomona College Abstract In spite of the proveable rarity of integer matrices with integer eigenvalues, they are commonly used as examples in introductory courses. We present a quick method for constructing such matrices starting with a given set of eigenvectors. The main feature of the method is an added level of flexibility in the choice of allowable eigenvalues. The method is also applicable to non-diagonalizable matrices, when given a basis of generalized eigenvectors. We have produced an online web tool that implements these constructions. Keywords: Integer matrices, Integer eigenvalues 2010 Mathematics Subject Classification: Primary 15A36; 15A18 Secondary 11C20 In this paper we will look at the problem of constructing a good problem. Most linear algebra and introductory ordinary differential equations classes include the topic of diagonalizing matrices: given a square matrix, finding its eigenvalues and constructing a basis of eigenvectors. In the instructional setting of such classes, concrete “toy" examples are helpful and perhaps even necessary (at least for most students). The examples that are typically given to students are, of course, integer-entry matrices with integer eigenvalues. Sometimes the eigenvalues are repeated with multipicity, sometimes they are all distinct. Oftentimes, the number 0 is avoided as an eigenvalue due to the degenerate cases it produces, particularly when the matrix in question comes from a linear system of differential equations. Yet in [10], Martin and Wong show that “Almost all integer matrices have no integer eigenvalues," let alone all integer eigenvalues. -
Fast Computation of the Rank Profile Matrix and the Generalized Bruhat Decomposition Jean-Guillaume Dumas, Clement Pernet, Ziad Sultan
Fast Computation of the Rank Profile Matrix and the Generalized Bruhat Decomposition Jean-Guillaume Dumas, Clement Pernet, Ziad Sultan To cite this version: Jean-Guillaume Dumas, Clement Pernet, Ziad Sultan. Fast Computation of the Rank Profile Matrix and the Generalized Bruhat Decomposition. Journal of Symbolic Computation, Elsevier, 2017, Special issue on ISSAC’15, 83, pp.187-210. 10.1016/j.jsc.2016.11.011. hal-01251223v1 HAL Id: hal-01251223 https://hal.archives-ouvertes.fr/hal-01251223v1 Submitted on 5 Jan 2016 (v1), last revised 14 May 2018 (v2) 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. Fast Computation of the Rank Profile Matrix and the Generalized Bruhat Decomposition Jean-Guillaume Dumas Universit´eGrenoble Alpes, Laboratoire LJK, umr CNRS, BP53X, 51, av. des Math´ematiques, F38041 Grenoble, France Cl´ement Pernet Universit´eGrenoble Alpes, Laboratoire de l’Informatique du Parall´elisme, Universit´ede Lyon, France. Ziad Sultan Universit´eGrenoble Alpes, Laboratoire LJK and LIG, Inria, CNRS, Inovall´ee, 655, av. de l’Europe, F38334 St Ismier Cedex, France Abstract The row (resp. column) rank profile of a matrix describes the stair-case shape of its row (resp. -
Ratner's Work on Unipotent Flows and Impact
Ratner’s Work on Unipotent Flows and Its Impact Elon Lindenstrauss, Peter Sarnak, and Amie Wilkinson Dani above. As the name suggests, these theorems assert that the closures, as well as related features, of the orbits of such flows are very restricted (rigid). As such they provide a fundamental and powerful tool for problems connected with these flows. The brilliant techniques that Ratner in- troduced and developed in establishing this rigidity have been the blueprint for similar rigidity theorems that have been proved more recently in other contexts. We begin by describing the setup for the group of 푑×푑 matrices with real entries and determinant equal to 1 — that is, SL(푑, ℝ). An element 푔 ∈ SL(푑, ℝ) is unipotent if 푔−1 is a nilpotent matrix (we use 1 to denote the identity element in 퐺), and we will say a group 푈 < 퐺 is unipotent if every element of 푈 is unipotent. Connected unipotent subgroups of SL(푑, ℝ), in particular one-parameter unipo- Ratner presenting her rigidity theorems in a plenary tent subgroups, are basic objects in Ratner’s work. A unipo- address to the 1994 ICM, Zurich. tent group is said to be a one-parameter unipotent group if there is a surjective homomorphism defined by polyno- In this note we delve a bit more into Ratner’s rigidity theo- mials from the additive group of real numbers onto the rems for unipotent flows and highlight some of their strik- group; for instance ing applications, expanding on the outline presented by Elon Lindenstrauss is Alice Kusiel and Kurt Vorreuter professor of mathemat- 1 푡 푡2/2 1 푡 ics at The Hebrew University of Jerusalem. -
Integer Matrix Approximation and Data Mining
Noname manuscript No. (will be inserted by the editor) Integer Matrix Approximation and Data Mining Bo Dong · Matthew M. Lin · Haesun Park Received: date / Accepted: date Abstract Integer datasets frequently appear in many applications in science and en- gineering. To analyze these datasets, we consider an integer matrix approximation technique that can preserve the original dataset characteristics. Because integers are discrete in nature, to the best of our knowledge, no previously proposed technique de- veloped for real numbers can be successfully applied. In this study, we first conduct a thorough review of current algorithms that can solve integer least squares problems, and then we develop an alternative least square method based on an integer least squares estimation to obtain the integer approximation of the integer matrices. We discuss numerical applications for the approximation of randomly generated integer matrices as well as studies of association rule mining, cluster analysis, and pattern extraction. Our computed results suggest that our proposed method can calculate a more accurate solution for discrete datasets than other existing methods. Keywords Data mining · Matrix factorization · Integer least squares problem · Clustering · Association rule · Pattern extraction The first author's research was supported in part by the National Natural Science Foundation of China under grant 11101067 and the Fundamental Research Funds for the Central Universities. The second author's research was supported in part by the National Center for Theoretical Sciences of Taiwan and by the Ministry of Science and Technology of Taiwan under grants 104-2115-M-006-017-MY3 and 105-2634-E-002-001. The third author's research was supported in part by the Defense Advanced Research Projects Agency (DARPA) XDATA program grant FA8750-12-2-0309 and NSF grants CCF-0808863, IIS-1242304, and IIS-1231742. -
Alternating Sign Matrices, Extensions and Related Cones
See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/311671190 Alternating sign matrices, extensions and related cones Article in Advances in Applied Mathematics · May 2017 DOI: 10.1016/j.aam.2016.12.001 CITATIONS READS 0 29 2 authors: Richard A. Brualdi Geir Dahl University of Wisconsin–Madison University of Oslo 252 PUBLICATIONS 3,815 CITATIONS 102 PUBLICATIONS 1,032 CITATIONS SEE PROFILE SEE PROFILE Some of the authors of this publication are also working on these related projects: Combinatorial matrix theory; alternating sign matrices View project All content following this page was uploaded by Geir Dahl on 16 December 2016. The user has requested enhancement of the downloaded file. All in-text references underlined in blue are added to the original document and are linked to publications on ResearchGate, letting you access and read them immediately. Alternating sign matrices, extensions and related cones Richard A. Brualdi∗ Geir Dahly December 1, 2016 Abstract An alternating sign matrix, or ASM, is a (0; ±1)-matrix where the nonzero entries in each row and column alternate in sign, and where each row and column sum is 1. We study the convex cone generated by ASMs of order n, called the ASM cone, as well as several related cones and polytopes. Some decomposition results are shown, and we find a minimal Hilbert basis of the ASM cone. The notion of (±1)-doubly stochastic matrices and a generalization of ASMs are introduced and various properties are shown. For instance, we give a new short proof of the linear characterization of the ASM polytope, in fact for a more general polytope. -
On the Computation of the HNF of a Module Over the Ring of Integers of A
On the computation of the HNF of a module over the ring of integers of a number field Jean-Fran¸cois Biasse Department of Mathematics and Statistics University of South Florida Claus Fieker Fachbereich Mathematik Universit¨at Kaiserslautern Postfach 3049 67653 Kaiserslautern - Germany Tommy Hofmann Fachbereich Mathematik Universit¨at Kaiserslautern Postfach 3049 67653 Kaiserslautern - Germany Abstract We present a variation of the modular algorithm for computing the Hermite normal form of an K -module presented by Cohen [7], where K is the ring of integers of a numberO field K. An approach presented in [7] basedO on reductions modulo ideals was conjectured to run in polynomial time by Cohen, but so far, no such proof was available in the literature. In this paper, we present a modification of the approach of [7] to prevent the coefficient swell and we rigorously assess its complexity with respect to the size of the input and the invariants of the field K. Keywords: Number field, Dedekind domain, Hermite normal form, relative extension of number fields arXiv:1612.09428v1 [math.NT] 30 Dec 2016 2000 MSC: 11Y40 1. Introduction Algorithms for modules over the rational integers such as the Hermite nor- mal form algorithm are at the core of all methods for computations with rings Email addresses: [email protected] (Jean-Fran¸cois Biasse), [email protected] (Claus Fieker), [email protected] (Tommy Hofmann) Preprint submitted to Elsevier April 22, 2018 and ideals in finite extensions of the rational numbers. Following the growing interest in relative extensions, that is, finite extensions of number fields, the structure of modules over Dedekind domains became important. -
K-Theory of Furstenberg Transformation Group C ∗-Algebras
Canad. J. Math. Vol. 65 (6), 2013 pp. 1287–1319 http://dx.doi.org/10.4153/CJM-2013-022-x c Canadian Mathematical Society 2013 K-theory of Furstenberg Transformation Group C∗-algebras Kamran Reihani Abstract. This paper studies the K-theoretic invariants of the crossed product C∗-algebras associated n with an important family of homeomorphisms of the tori T called Furstenberg transformations. Using the Pimsner–Voiculescu theorem, we prove that given n, the K-groups of those crossed products whose corresponding n × n integer matrices are unipotent of maximal degree always have the same rank an. We show using the theory developed here that a claim made in the literature about the torsion subgroups of these K-groups is false. Using the representation theory of the simple Lie algebra sl(2; C), we show that, remarkably, an has a combinatorial significance. For example, every a2n+1 is just the number of ways that 0 can be represented as a sum of integers between −n and n (with no repetitions). By adapting an argument of van Lint (in which he answered a question of Erdos),˝ a simple explicit formula for the asymptotic behavior of the sequence fang is given. Finally, we describe the order structure of the K0-groups of an important class of Furstenberg crossed products, obtaining their complete Elliott invariant using classification results of H. Lin and N. C. Phillips. 1 Introduction Furstenberg transformations were introduced in [9] as the first examples of homeo- morphisms of the tori that under some necessary and sufficient conditions are mini- mal and uniquely ergodic. -
Package 'Matrixcalc'
Package ‘matrixcalc’ July 28, 2021 Version 1.0-5 Date 2021-07-27 Title Collection of Functions for Matrix Calculations Author Frederick Novomestky <[email protected]> Maintainer S. Thomas Kelly <[email protected]> Depends R (>= 2.0.1) Description A collection of functions to support matrix calculations for probability, econometric and numerical analysis. There are additional functions that are comparable to APL functions which are useful for actuarial models such as pension mathematics. This package is used for teaching and research purposes at the Department of Finance and Risk Engineering, New York University, Polytechnic Institute, Brooklyn, NY 11201. Horn, R.A. (1990) Matrix Analysis. ISBN 978-0521386326. Lancaster, P. (1969) Theory of Matrices. ISBN 978-0124355507. Lay, D.C. (1995) Linear Algebra: And Its Applications. ISBN 978-0201845563. License GPL (>= 2) Repository CRAN Date/Publication 2021-07-28 08:00:02 UTC NeedsCompilation no R topics documented: commutation.matrix . .3 creation.matrix . .4 D.matrix . .5 direct.prod . .6 direct.sum . .7 duplication.matrix . .8 E.matrices . .9 elimination.matrix . 10 entrywise.norm . 11 1 2 R topics documented: fibonacci.matrix . 12 frobenius.matrix . 13 frobenius.norm . 14 frobenius.prod . 15 H.matrices . 17 hadamard.prod . 18 hankel.matrix . 19 hilbert.matrix . 20 hilbert.schmidt.norm . 21 inf.norm . 22 is.diagonal.matrix . 23 is.idempotent.matrix . 24 is.indefinite . 25 is.negative.definite . 26 is.negative.semi.definite . 28 is.non.singular.matrix . 29 is.positive.definite . 31 is.positive.semi.definite . 32 is.singular.matrix . 34 is.skew.symmetric.matrix . 35 is.square.matrix . 36 is.symmetric.matrix . -
Generalized Spectral Characterizations of Almost Controllable Graphs
Generalized spectral characterizations of almost controllable graphs Wei Wanga Fenjin Liub∗ Wei Wangc aSchool of Mathematics, Physics and Finance, Anhui Polytechnic University, Wuhu 241000, P.R. China bSchool of Science, Chang’an University, Xi’an 710064, P.R. China bSchool of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, P.R. China Abstract Characterizing graphs by their spectra is an important topic in spectral graph theory, which has attracted a lot of attention of researchers in recent years. It is generally very hard and challenging to show a given graph to be determined by its spectrum. In Wang [J. Combin. Theory, Ser. B, 122 (2017): 438-451], the author gave a simple arithmetic condition for a family of graphs being determined by their generalized spectra. However, the method applies only to a family of the so called controllable graphs; it fails when the graphs are non-controllable. In this paper, we introduce a class of non-controllable graphs, called almost con- trollable graphs, and prove that, for any pair of almost controllable graphs G and H that are generalized cospectral, there exist exactly two rational orthogonal matrices Q with constant row sums such that QTA(G)Q = A(H), where A(G) and A(H) are the adjacency matrices of G and H, respectively. The main ingredient of the proof is a use of the Binet-Cauchy formula. As an application, we obtain a simple criterion for an almost controllable graph G to be determined by its generalized spectrum, which in some sense extends the corresponding result for controllable graphs. -
Computing Rational Forms of Integer Matrices
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector J. Symbolic Computation (2002) 34, 157–172 doi:10.1006/jsco.2002.0554 Available online at http://www.idealibrary.com on Computing Rational Forms of Integer Matrices MARK GIESBRECHT† AND ARNE STORJOHANN‡ School of Computer Science, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1 n×n A new algorithm is presented for finding the Frobenius rational form F ∈ Z of any n×n 4 3 A ∈ Z which requires an expected number of O(n (log n + log kAk) + n (log n + log kAk)2) word operations using standard integer and matrix arithmetic (where kAk = maxij |Aij |). This substantially improves on the fastest previously known algorithms. The algorithm is probabilistic of the Las Vegas type: it assumes a source of random bits but always produces the correct answer. Las Vegas algorithms are also presented for computing a transformation matrix to the Frobenius form, and for computing the rational Jordan form of an integer matrix. c 2002 Elsevier Science Ltd. All rights reserved. 1. Introduction In this paper we present new algorithms for exactly computing the Frobenius and rational Jordan normal forms of an integer matrix which are substantially faster than those previously known. We show that the Frobenius form F ∈ Zn×n of any A ∈ Zn×n can be computed with an expected number of O(n4(log n+log kAk)+n3(log n+log kAk)2) word operations using standard integer and matrix arithmetic. Here and throughout this paper, kAk = maxij |Aij|. -
An Application of the Hermite Normal Form in Integer Programming
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector An Application of the Hermite Normal Form in Integer Programming Ming S. Hung Graduate School of Management Kent State University Kent, Ohio 44242 and Walter 0. Rom School of Business Administration Cleveland State University Cleveland, Ohio 44115 Submitted by Richard A. Brualdi ABSTRACT A new algorithm for solving integer programming problems is developed. It is based on cuts which are generated from the Hermite normal form of the basis matrix at a continuous extreme point. INTRODUCTION We present a new cutting plane algorithm for solving an integer program. The cuts are derived from a modified version of the usual column tableau used in integer programming: Rather than doing a complete elimination on the rows corresponding to the nonbasic variables, these rows are transformed into the Her-mite normal form. There are two major advantages with our cutting plane algorithm: First, it uses all integer operations in the computa- tions. Second, the cuts are deep, since they expose vertices of the integer hull. The Her-mite normal form (HNF), along with the basic properties of the cuts, is developed in Section I. Sections II and III apply the HNF to systems of equations and integer programs. A proof of convergence for the algorithm is presented in Section IV. Section V discusses some preliminary computa- tional experience. LINEAR ALGEBRA AND ITS APPLlCATlONS 140:163-179 (1990) 163 0 Else&r Science Publishing Co., Inc., 1990 655 Avenue of the Americas, New York, NY 10010 0024-3795/90/$3.50 164 MING S.