DOCSLIB.ORG

On the Q-Analogue of Cauchy Matrices Alessandro Neri

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
On the Q-Analogue of Cauchy Matrices Alessandro Neri I am a friend of Finite Geometry! On the q-Analogue of Cauchy Matrices Alessandro Neri 17-21 June 2019 - VUB Alessandro Neri 19 June 2019 1 / 19 On the q-Analogue of Cauchy Matrices Alessandro Neri I am a friend of Finite Geometry! 17-21 June 2019 - VUB Alessandro Neri 19 June 2019 1 / 19 q-Analogues Model Finite set Finite dim vector space over Fq Element 1-dim subspace ; f0g Cardinality Dimension Intersection Intersection Union Sum Alessandro Neri 19 June 2019 1 / 19 Examples 1. Binomials and q-binomials: k−1 k−1 n Y n − i n Y 1 − qn−i = ; = : k k − i k 1 − qk−i i=0 q i=0 2. (Chu)-Vandermonde and q-Vandermonde identity: k k m + n X m n m + n X m n = ; = qj(m−k+j): k j k − j k k − j j j=0 q j=0 q q 3. Polynomials and q-polynomials: k q qk a0 + a1x + ::: + ak x ; a0x + a1x + ::: + ak x : 4. Gamma and q-Gamma functions: 1 − qx Γ(z + 1) = zΓ(z); Γ (x + 1) = Γ (x): q 1 − q q Alessandro Neri 19 June 2019 2 / 19 For n = k, det(V ) 6= 0 if and only if the αi 's are all distinct. jfα1; : : : ; αngj = n: In particular, all the k × k minors of V are non-zero. Vandermonde Matrix Let k ≤ n. 0 1 1 ::: 1 1 B α1 α2 : : : αn C B C B α2 α2 : : : α2 C V = B 1 2 n C B . C @ . A k−1 k−1 k−1 α1 α2 : : : αn Alessandro Neri 19 June 2019 3 / 19 Vandermonde Matrix Let k ≤ n. 0 1 1 ::: 1 1 B α1 α2 : : : αn C B C B α2 α2 : : : α2 C V = B 1 2 n C B . C @ . A k−1 k−1 k−1 α1 α2 : : : αn For n = k, det(V ) 6= 0 if and only if the αi 's are all distinct. jfα1; : : : ; αngj = n: In particular, all the k × k minors of V are non-zero. Alessandro Neri 19 June 2019 3 / 19 For n = k, det(G) 6= 0 if and only if the gi 's are all Fq-linearly independent. dimFq hg1;:::; gniFq = n: In particular, for every M 2 GLn(Fq), all the k × k minors of GM are non-zero. Moore Matrix Let k ≤ n. 0 g1 g2 ::: gn 1 q q q B g1 g2 ::: gn C B C G = B . C ; @ . A qk−1 qk−1 qk−1 g1 g2 ::: gn Alessandro Neri 19 June 2019 4 / 19 Moore Matrix Let k ≤ n. 0 g1 g2 ::: gn 1 q q q B g1 g2 ::: gn C B C G = B . C ; @ . A qk−1 qk−1 qk−1 g1 g2 ::: gn For n = k, det(G) 6= 0 if and only if the gi 's are all Fq-linearly independent. dimFq hg1;:::; gniFq = n: In particular, for every M 2 GLn(Fq), all the k × k minors of GM are non-zero. Alessandro Neri 19 June 2019 4 / 19 Moore Matrix Let k ≤ n, σ : x 7−! x q 0 1 g1 g2 ::: gn B σ(g1) σ(g2) : : : σ(gn) C G B C ; = B . C @ . A k−1 k−1 k−1 σ (g1) σ (g2) : : : σ (gn) For n = k, det(G) 6= 0 if and only if the gi 's are all Fq-linearly independent. dimFq hg1;:::; gniFq = n: In particular, for every M 2 GLn(Fq), all the k × k minors of GM are non-zero. Alessandro Neri 19 June 2019 5 / 19 Moore matrix 1. Dickson used the Moore matrix for finding the modular invariants of the general linear group over a finite field. 2. It is widely used in the study of normal bases of finite fields. 3. Y Y det(G) = (c1g1 + ··· + ci−1gi−1 + gi ) : 1≤i≤n c1;:::;ci−12Fq Alessandro Neri 19 June 2019 6 / 19 r×s The matrix X 2 Fq defined by ci dj Xi;j = xi − yj is called Generalized Cauchy (GC) Matrix. For j ≤ minfr; sg, all the j × j minors are non-zero. Generalized Cauchy Matrix (1) x1;:::; xr 2 Fq pairwise distinct, (2) y1;:::; ys 2 Fq pairwise distinct, (3) y1;:::; ys 2 Fq n fx1;:::; xr g, ∗ (4) c1;:::; cr ; d1;:::; ds 2 Fq. Alessandro Neri 19 June 2019 7 / 19 Generalized Cauchy Matrix (1) x1;:::; xr 2 Fq pairwise distinct, (2) y1;:::; ys 2 Fq pairwise distinct, (3) y1;:::; ys 2 Fq n fx1;:::; xr g, ∗ (4) c1;:::; cr ; d1;:::; ds 2 Fq. r×s The matrix X 2 Fq defined by ci dj Xi;j = xi − yj is called Generalized Cauchy (GC) Matrix. For j ≤ minfr; sg, all the j × j minors are non-zero. Alessandro Neri 19 June 2019 7 / 19 Question How to define a q-analogue of Cauchy matrices? Alessandro Neri 19 June 2019 8 / 19 α1; : : : ; αn 2 Fq distinct elements ∗ b1;:::; bn 2 Fq C = f(b1f (α1); b2f (α2);:::; bnf (αn)) j f 2 Fq[x]<k g THEN C is the Generalized Reed-Solomon code GRSn;k (α; b) Generalized Reed-Solomon Codes [Reed, Solomon '60] Fq[x]<k := ff (x) 2 Fq[x] j deg f < kg : Alessandro Neri 19 June 2019 9 / 19 THEN C is the Generalized Reed-Solomon code GRSn;k (α; b) Generalized Reed-Solomon Codes [Reed, Solomon '60] Fq[x]<k := ff (x) 2 Fq[x] j deg f < kg : α1; : : : ; αn 2 Fq distinct elements ∗ b1;:::; bn 2 Fq C = f(b1f (α1); b2f (α2);:::; bnf (αn)) j f 2 Fq[x]<k g Alessandro Neri 19 June 2019 9 / 19 Generalized Reed-Solomon Codes [Reed, Solomon '60] Fq[x]<k := ff (x) 2 Fq[x] j deg f < kg : α1; : : : ; αn 2 Fq distinct elements ∗ b1;:::; bn 2 Fq C = f(b1f (α1); b2f (α2);:::; bnf (αn)) j f 2 Fq[x]<k g THEN C is the Generalized Reed-Solomon code GRSn;k (α; b) Alessandro Neri 19 June 2019 9 / 19 Generalized Reed-Solomon Codes [Reed, Solomon '60] Fq[x]<k := ff (x) 2 Fq[x] j deg f < kg : α1; : : : ; αn 2 Fq s.t. jfα1; : : : ; αngj = n ∗ b1;:::; bn 2 Fq C = f(b1f (α1); b2f (α2);:::; bnf (αn)) j f 2 Fq[x]<k g THEN C is the Generalized Reed-Solomon code GRSn;k (α; b) Alessandro Neri 19 June 2019 9 / 19 Canonical Generator Matrix for GRS codes Consider the canonical monomial basis f1; x;:::; x k−1g and evaluate it. We get the following generator matrix Weighted Vandermonde (WV) matrix 0 1 1 ::: 1 1 B α1 α2 : : : αn C B C B α2 α2 : : : α2 C B 1 2 n C diag(b) B . C @ . A k−1 k−1 k−1 α1 α2 : : : αn Alessandro Neri 19 June 2019 10 / 19 3. If C is MDS the generator matrix in RREF is of the form (Ik j X ). 4. GRS codes are MDS. Theorem [Roth, Seroussi '85] There is a 1-1 correspondence between GC matrices and GRS codes codes given by X ! rowsp( Ik j X ): GRS Codes and Cauchy Matrices k×n 1. Every [n; k]q code has many generator matrices G 2 Fq . 2. Every [n; k]q code has a unique generator matrix in Reduced Row Echelon Form (RREF). Alessandro Neri 19 June 2019 11 / 19 Theorem [Roth, Seroussi '85] There is a 1-1 correspondence between GC matrices and GRS codes codes given by X ! rowsp( Ik j X ): GRS Codes and Cauchy Matrices k×n 1. Every [n; k]q code has many generator matrices G 2 Fq . 2. Every [n; k]q code has a unique generator matrix in Reduced Row Echelon Form (RREF). 3. If C is MDS the generator matrix in RREF is of the form (Ik j X ). 4. GRS codes are MDS. Alessandro Neri 19 June 2019 11 / 19 Theorem [Roth, Seroussi '85] There is a 1-1 correspondence between GC matrices and GRS codes codes given by X ! rowsp( Ik j X ): GRS Codes and Cauchy Matrices k×n 1. Every [n; k]q code has many generator matrices G 2 Fq . 2. Every [n; k]q code has a unique generator matrix in Reduced Row Echelon Form (RREF). 3. If C is MDS the generator matrix in RREF is of the form (Ik j X ). 4. GRS codes are MDS. Alessandro Neri 19 June 2019 11 / 19 GRS Codes and Cauchy Matrices k×n 1. Every [n; k]q code has many generator matrices G 2 Fq . 2. Every [n; k]q code has a unique generator matrix in Reduced Row Echelon Form (RREF). 3. If C is MDS the generator matrix in RREF is of the form (Ik j X ). 4. GRS codes are MDS. Theorem [Roth, Seroussi '85] There is a 1-1 correspondence between GC matrices and GRS codes codes given by X ! rowsp( Ik j X ): Alessandro Neri 19 June 2019 11 / 19 Rank Distance m×n The rank distance dR on Fq is defined by m×n dR (X ; Y ) := rk(X − Y ); X ; Y 2 Fq : n The rank distance dR on Fqm is defined by dR (u; v) := dimhu1 − v1; u2 − v2;:::; un − vniFq : n A rank metric code C is a subset of Fqm equipped with the rank distance. General Setting Fqm extension field of degree m of a finite field Fq. ∼ m $ Fqm = Fq as vector spaces over Fq. n ∼ m×n $ Fqm = Fq as vector spaces over Fq.
Load more

Recommended publications
  • Paper We Shall Assume That All Logs Are Base 2
    Maximally Recoverable Codes: the Bounded Case Venkata Gandikota ∗ Elena Grigorescu y Clayton Thomas z Minshen Zhu x Abstract Modern distributed storage systems employ Maximally Recoverable codes that aim to bal- ance failure recovery capabilities with encoding/decoding efficiency tradeoffs. Recent works of Gopalan et al [SODA 2017] and Kane et al [FOCS 2017] show that the alphabet size of grid-like topologies of practical interest must be large, a feature that hampers decoding efficiency. To bypass such shortcomings, in this work we initiate the study of a weaker version of recoverability, where instead of being able to correct all correctable erasure patterns (as is the case for maximal recoverability), we only require to correct all erasure patterns of bounded size. The study of this notion reduces to a variant of a combinatorial problem studied in the literature, which is interesting in its own right. We study the alphabet size of codes withstanding all erasure patterns of small (constant) size. We believe the questions we propose are relevant to both real storage systems and combinatorial analysis, and merit further study. 1 Introduction Modern distributed storage systems need to address the challenge of storing large amounts of data, with small overhead, while providing reliable recovery in the face of failures. Unlike in com- munication settings, if a failure occurs in a storage system, it is typically easy to detect where it occurs (e.g., rack or data center failure are obvious to the system). Hence, recent trends in practical systems adopt erasure coding schemes with fast encoding and decoding capabilities [BHH13, HSX+12, SLR+14].
    [Show full text]
  • Seminar VII for the Course GROUP THEORY in PHYSICS Micael Flohr
    Seminar VII for the course GROUP THEORY IN PHYSICS Mic~ael Flohr The classical Lie groups 25. January 2005 MATRIX LIE GROUPS Most Lie groups one ever encouters in physics are realized as matrix Lie groups and thus as subgroups of GL(n, R) or GL(n, C). This is the group of invertibel n × n matrices with coefficients in R or C, respectively. This is a Lie group, since it forms as an open subset of the vector space of n × n matrices a manifold. Matrix multiplication is certainly a differentiable map, as is taking the inverse via Cramer’s rule. The only condition defining the open 2 subset is that the determinat must not be zero, which implies that dimKGL(n, K) = n is the same as the one of the vector space Mn(K). However, GL(n, R) is not connected, because we cannot move continuously from a matrix with determinant less than zero to one with determinant larger than zero. It is worth mentioning that gl(n, K) is the vector space of all n × n matrices over the field K, equipped with the standard commutator as Lie bracket. We can describe most other Lie groups as subgroups of GL(n, K) for either K = R or K = C. There are two ways to do so. Firstly, one can give restricting equations to the coefficients of the matrices. Secondly, one can find subgroups of the automorphisms of V =∼ Kn, which conserve a given structure on Kn. In the following, we give some examples for this: SL(n, K).
    [Show full text]
  • Inertia of the Matrix [(Pi + Pj) ]
    isid/ms/2013/12 October 20, 2013 http://www.isid.ac.in/estatmath/eprints r Inertia of the matrix [(pi + pj) ] Rajendra Bhatia and Tanvi Jain Indian Statistical Institute, Delhi Centre 7, SJSS Marg, New Delhi{110 016, India r INERTIA OF THE MATRIX [(pi + pj) ] RAJENDRA BHATIA* AND TANVI JAIN** Abstract. Let p1; : : : ; pn be positive real numbers. It is well r known that for every r < 0 the matrix [(pi + pj) ] is positive def- inite. Our main theorem gives a count of the number of positive and negative eigenvalues of this matrix when r > 0: Connections with some other matrices that arise in Loewner's theory of oper- ator monotone functions and in the theory of spline interpolation are discussed. 1. Introduction Let p1; p2; : : : ; pn be distinct positive real numbers. The n×n matrix 1 C = [ ] is known as the Cauchy matrix. The special case pi = i pi+pj 1 gives the Hilbert matrix H = [ i+j ]: Both matrices have been studied by several authors in diverse contexts and are much used as test matrices in numerical analysis. The Cauchy matrix is known to be positive definite. It possessesh ai ◦r 1 stronger property: for each r > 0 the entrywise power C = r (pi+pj ) is positive definite. (See [4] for a proof.) The object of this paper is to study positivity properties of the related family of matrices r Pr = [(pi + pj) ]; r ≥ 0: (1) The inertia of a Hermitian matrix A is the triple In(A) = (π(A); ζ(A); ν(A)) ; in which π(A); ζ(A) and ν(A) stand for the number of positive, zero, and negative eigenvalues of A; respectively.
    [Show full text]
  • Hermitian, Symmetric and Symplectic Random Ensembles
    Annals of Mathematics, 153 (2001), 149–189 Hermitian, symmetric and symplectic random ensembles: PDEs for the distribution of the spectrum By M. Adler and P. van Moerbeke* Abstract Given the Hermitian, symmetric and symplectic ensembles, it is shown that the probability that the spectrum belongs to one or several intervals sat- isfies a nonlinear PDE. This is done for the three classical ensembles: Gaussian, Laguerre and Jacobi. For the Hermitian ensemble, the PDE (in the boundary points of the intervals) is related to the Toda lattice and the KP equation, whereas for the symmetric and symplectic ensembles the PDE is an inductive equation, related to the so-called Pfaff-KP equation and the Pfaff lattice. The method consists of inserting time-variables in the integral and showing that this integral satisfies integrable lattice equations and Virasoro constraints. Contents 0. Introduction 0.1. Hermitian, symmetric and symplectic Gaussian ensembles 0.2. Hermitian, symmetric and symplectic Laguerre ensembles 0.3. Hermitian, symmetric and symplectic Jacobi ensembles 0.4. ODEs, when E has one boundary point 1. Beta-integrals 1.1. Virasoro constraints for β-integrals 1.2. Proof: β-integrals as fixed points of vertex operators 1.3. Examples arXiv:math-ph/0009001v2 14 Aug 2001 2. Matrix integrals and associated integrable systems 2.1. Hermitian matrix integrals and the Toda lattice 2.2. Symmetric/symplectic matrix integrals and the Pfaff lattice 3. Expressing t-partials in terms of boundary-partials 3.1. Gaussian and Laguerre ensembles 3.2. Jacobi ensemble 3.3. Evaluating the matrix integrals on the full range ∗The support of a National Science Foundation grant #DMS-98-4-50790 is gratefully acknowl- edged.
    [Show full text]
  • Fast Approximation Algorithms for Cauchy Matrices, Polynomials and Rational Functions
    City University of New York (CUNY) CUNY Academic Works Computer Science Technical Reports CUNY Academic Works 2013 TR-2013011: Fast Approximation Algorithms for Cauchy Matrices, Polynomials and Rational Functions Victor Y. Pan How does access to this work benefit ou?y Let us know! More information about this work at: https://academicworks.cuny.edu/gc_cs_tr/386 Discover additional works at: https://academicworks.cuny.edu This work is made publicly available by the City University of New York (CUNY). Contact: [email protected] Fast Approximation Algorithms for Cauchy Matrices, Polynomials and Rational Functions ? Victor Y. Pan Department of Mathematics and Computer Science Lehman College and the Graduate Center of the City University of New York Bronx, NY 10468 USA [email protected], home page: http://comet.lehman.cuny.edu/vpan/ Abstract. The papers [MRT05], [CGS07], [XXG12], and [XXCBa] have combined the advanced FMM techniques with transformations of matrix structures (traced back to [P90]) in order to devise numerically stable algorithms that approximate the solutions of Toeplitz, Hankel, Toeplitz- like, and Hankel-like linear systems of equations in nearly linear arith- metic time, versus classical cubic time and quadratic time of the previous advanced algorithms. We show that the power of these approximation al- gorithms can be extended to yield similar results for computations with other matrices that have displacement structure, which includes Van- dermonde and Cauchy matrices, as well as to polynomial and rational evaluation and interpolation. The resulting decrease of the running time of the known approximation algorithms is again by order of magnitude, from quadratic to nearly linear.
    [Show full text]
  • An Accurate Computational Solution of Totally Positive Cauchy Linear Systems
    AN ACCURATE COMPUTATIONAL SOLUTION OF TOTALLY POSITIVE CAUCHY LINEAR SYSTEMS Abdoulaye Cissé Andrew Anda Computer Science Department Computer Science Department St. Cloud State University St. Cloud State University St. Cloud, MN 56301 St. Cloud, MN 56301 [email protected] [email protected] Abstract ≈ ’ ∆ 1 The Cauchy linear systems C(x, y)V = b where C(x, y) = ∆ , x = (xi ) , y = ( yi ) , « xi − y j V = (vi ) , b = (bi ) can be solved very accurately regardless of condition number using Björck-Pereyra-type methods. We explain this phenomenon by the following facts. First, the Björck-Pereyra methods, written in matrix form, are essentially an accurate and efficient bidiagonal decomposition of the inverse of the matrix, as obtained by Neville Elimination with no pivoting. Second, each nontrivial entry of this bidiagonal decomposition is a product of two quotients of (initial) minors. We use this method to derive and implement a new method of accurately and efficiently solving totally positive Cauchy linear systems in time complexity O(n2). Keywords: Accuracy, Björck-Pereyra-type methods, Cauchy linear systems, floating point architectures, generalized Vandermonde linear systems, Neville elimination, totally positive, LDU decomposition. Note: In this paper we assume that the nodes xi and yj are pair wise distinct and C is totally positive 1 Background Definition 1.1 A Cauchy matrix is defined as: ≈ ’ ∆ 1 C(x, y) = ∆ , for i, j = 1,2,..., n (1.1) « xi − y j where the nodes xi and yj are assumed pair wise distinct. C(x,y) is totally positive (TP) if 2 2 0 < y1 < y2 < < yn−1 < yn < x1 < x2 < < xn−1 < xn .
    [Show full text]
  • Displacement Rank of the Drazin Inverse Huaian Diaoa, Yimin Weib;1, Sanzheng Qiaoc;∗;2
    Available online at www.sciencedirect.com Journal of Computational and Applied Mathematics 167 (2004) 147–161 www.elsevier.com/locate/cam Displacement rank of the Drazin inverse Huaian Diaoa, Yimin Weib;1, Sanzheng Qiaoc;∗;2 aInstitute of Mathematics, Fudan University, Shanghai 200433, PR China bDepartment of Mathematics, Fudan University, Shanghai 200433, PR China cDepartment of Computing and Software, McMaster University, Hamilton, Ont., Canada L8S 4L7 Received 20 February 2003; received in revised form 4 August 2003 Abstract In this paper, we study the displacement rank of the Drazin inverse. Both Sylvester displacement and the generalized displacement are discussed. We present upper bounds for the ranks of the displacements of the Drazin inverse. The general results are applied to the group inverse of a structured matrix such as close-to- Toeplitz, generalized Cauchy, Toeplitz-plus-Hankel, and Bezoutians. c 2003 Elsevier B.V. All rights reserved. MSC: 15A09; 65F20 Keywords: Drazin inverse; Displacement; Structured matrix; Jordan canonical form 1. Introduction Displacement gives a quantitative way of identifying the structure of a matrix. Consider an n × n Toeplitz matrix t0 t−1 ··· t−n+1 . t .. .. 1 T = ; . .. .. . t−1 tn−1 ··· t1 t0 ∗ Corresponding author. Tel.: +1-905-525-9140; fax: +1-905-524-0340. E-mail addresses: [email protected] (Y. Wei), [email protected] (S. Qiao). 1 Supported by National Natural Science Foundation of China under Grant 19901006. 2 Partially supported by Natural Science and Engineering Research Council of Canada. 0377-0427/$ - see front matter c 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.cam.2003.09.050 148 H.
    [Show full text]
  • On Decoding Additive Generalized Twisted Gabidulin Codes
    Cryptography and Communications (2020) 12:987–1009 https://doi.org/10.1007/s12095-020-00449-9 On decoding additive generalized twisted Gabidulin codes Wrya K. Kadir1 · Chunlei Li1 Received: 20 September 2019 / Accepted: 30 June 2020 /Published online: 23 July 2020 © The Author(s) 2020 Abstract In this paper, we consider an interpolation-based decoding algorithm for a large family of maximum rank distance codes, known as the additive generalized twisted Gabidulin codes, over the finite field Fqn for any prime power q. This paper extends the work of the con- ference paper Li and Kadir (2019) presented at the International Workshop on Coding and Cryptography 2019, which decoded these codes over finite fields in characteristic two. Keywords Rank metric · Maximum rank distance codes · Gabidulin codes · Twisted Gabidulin codes · Generalized twisted Gabidulin codes Mathematics Subject Classification (2010) 94B35 · 68P30 · 11T71 · 11T06 1 Introduction Error correction codes with the rank metric have found applications in space-time coding [27], random network coding [44] and cryptography [12]. Many important properties of rank metric codes including the Singleton like bound were independently studied by Del- sarte [9] , Gabidulin [13] and Roth [38]. Codes that achieve this bound were called maximum rank distance (MRD) codes. The most famous sub-family of MRD codes are Gabidulin codes which is the rank metric analog of Reed-Solomon codes. They have been extensively studied in the literature [9, 12, 13, 25, 36, 38]. Finding new families of MRD codes has been an interesting research topic since the invention of Gabidulin codes. In [20, 39], the Frobenious automorphism in the Gabidulin codes were generalized to arbitrary automorphism and generalized Gabidulin (GG) codes Chunlei Li [email protected] Wrya K.
    [Show full text]
  • The Geometry of Rank Decompositions of Matrix Multiplication Ii: 3 3 Matrices
    THE GEOMETRY OF RANK DECOMPOSITIONS OF MATRIX MULTIPLICATION II: 3 3 MATRICES × GREY BALLARD, CHRISTIAN IKENMEYER, J.M. LANDSBERG, AND NICK RYDER Abstract. This is the second in a series of papers on rank decompositions of the matrix multiplication tensor. We present new rank 23 decompositions for the 3×3 matrix multiplication tensor M⟨3⟩. All our decompositions have symmetry groups that include the standard cyclic permutation of factors but otherwise exhibit a range of behavior. One of them has 11 cubes as summands and admits an unexpected symmetry group of order 12. We establish basic information regarding symmetry groups of decompositions and outline two approaches for finding new rank decompositions of M⟨n⟩ for larger n. 1. Introduction This is the second in a planned series of papers on the geometry of rank decompositions n2 n2 n2 of the matrix multiplication tensor M n C C C . Our goal is to obtain new rank m m m decompositions of M n by exploiting symmetry. For a tensor T C C C , the rank of r ⟨ ⟩ m T is the smallest r such that T j 1 aj ∈bj cj⊗, with⊗ aj; bj; cj C . The rank of M n is a standard complexity measure⟨ ⟩ of matrix multiplication, in particular,∈ it governs⊗ ⊗ the total number of arithmetic operations needed to= multiply∑ = ⊗ two⊗ matrices. ∈ ⟨ ⟩ In this paper we present rank 23 decompositions of M 3 that have large symmetry groups, in particular all admit the standard cyclic Z3-symmetry of permuting the three tensor factors. ⟨ ⟩ Although many rank 23 decompositions of M 3 are known [15, 13], none of them was known to admit the standard cyclic Z3-symmetry.
    [Show full text]
  • Arxiv:2104.06123V2 [Nlin.SI] 19 Jul 2021
    LINEAR INTEGRAL EQUATIONS AND TWO-DIMENSIONAL TODA SYSTEMS YUE YIN AND WEI FU Abstract. The direct linearisation framework is presented for the two-dimensional Toda equations associated with the infinite-dimensional (1) (2) (1) (2) Z+ Lie algebras A∞, B∞ and C∞, as well as the Kac–Moody algebras Ar , A2r , Cr and Dr+1 for arbitrary integers r ∈ , from the aspect of a set of linear integral equations in a certain form. Such a scheme not only provides a unified perspective to understand the underlying integrability structure, but also induces the direct linearising type solution potentially leading to the universal solution space, for each class of the two-dimensional Toda system. As particular applications of this framework to the two-dimensional Toda lattices, we rediscover the Lax pairs and the adjoint Lax pairs and simultaneously construct the generalised Cauchy matrix solutions. 1. Introduction In the modern theory of integrable systems, the notion of integrability of nonlinear equations often refers to the property that a differential/difference equation is exactly solvable under an initial-boundary condition, which in many cases allows us to construct explicit solutions. Motivated by this, many mathematical methods were invented and developed to search for explicit solutions of nonlinear models, for instance, the inverse scattering transform, the Darboux transform, Hirota’s bilinear method, as well as the algebro-geometric method, etc., see e.g. the monographs [1,11,16,26]. These techniques not only explained the nonlinear phenomena such as solitary waves and periodic waves in nature mathematically, but also motivated the discoveryof a huge number of nonlinear equations that possess “nice” algebraic and geometric properties.
    [Show full text]
  • Fast Approximate Computations with Cauchy Matrices and Polynomials ∗
    Fast Approximate Computations with Cauchy Matrices and Polynomials ∗ Victor Y. Pan Departments of Mathematics and Computer Science Lehman College and the Graduate Center of the City University of New York Bronx, NY 10468 USA [email protected] http://comet.lehman.cuny.edu/vpan/ Abstract Multipoint polynomial evaluation and interpolation are fundamental for modern symbolic and nu- merical computing. The known algorithms solve both problems over any field of constants in nearly linear arithmetic time, but the cost grows to quadratic for numerical solution. We fix this discrepancy: our new numerical algorithms run in nearly linear arithmetic time. At first we restate our goals as the multiplication of an n × n Vandermonde matrix by a vector and the solution of a Vandermonde linear system of n equations. Then we transform the matrix into a Cauchy structured matrix with some special features. By exploiting them, we approximate the matrix by a generalized hierarchically semiseparable matrix, which is a structured matrix of a different class. Finally we accelerate our solution to the original problems by applying Fast Multipole Method to the latter matrix. Our resulting numerical algorithms run in nearly optimal arithmetic time when they perform the above fundamental computations with polynomials, Vandermonde matrices, transposed Vandermonde matrices, and a large class of Cauchy and Cauchy-like matrices. Some of our techniques may be of independent interest. Key words: Polynomial evaluation; Rational evaluation; Interpolation; Vandermonde matrices; Transfor- mation of matrix structures; Cauchy matrices; Fast Multipole Method; HSS matrices; Matrix compression AMS Subject Classification: 12Y05, 15A04, 47A65, 65D05, 68Q25 1 Introduction 1.1 The background and our progress Multipoint polynomial evaluation and interpolation are fundamental for modern symbolic and numerical arXiv:1506.02285v3 [math.NA] 17 Apr 2017 computing.
    [Show full text]
  • Positivity Properties of the Matrix (I + J)I+J
    isid/ms/2015/11 September 21, 2015 http://www.isid.ac.in/∼statmath/index.php?module=Preprint Positivity properties of the matrix (i + j)i+j Rajendra Bhatia and Tanvi Jain Indian Statistical Institute, Delhi Centre 7, SJSS Marg, New Delhi{110 016, India POSITIVITY PROPERTIES OF THE MATRIX [(i + j)i+j] RAJENDRA BHATIA AND TANVI JAIN Abstract. Let p1 < p2 < ··· < pn be positive real numbers. pi+pj It is shown that the matrix whose i; j entry is (pi + pj) is infinitely divisible, nonsingular and totally positive. 1. Introduction Matrices whose entries are obtained by assembling natural numbers in special ways often possess interesting properties. The most famous h 1 i example of such a matrix is the Hilbert matrix H = i+j−1 which has inspired a lot of work in diverse areas. Some others are the min i+j matrix M = min(i; j) ; and the Pascal matrix P = i . There is a considerable body of literature around each of these matrices, a sample of which can be found in [3], [5] and [7]. In this note we initiate the study of one more matrix of this type. Let A be the n×n matrix with its (i; j) entry equal to (i+j −1)i+j−1. Thus 2 1 22 33 ··· nn 3 6 22 33 44 ··· (n + 1)n+1 7 6 3 4 5 7 A = 6 3 4 5 ······ 7 : (1) 6 7 4 ··············· 5 nn ········· (2n − 1)2n−1 More generally, let p1 < p2 < ··· < pn be positive real numbers, and consider the n × n matrix pi+pj B = (pi + pj) : (2) The special choice pi = i − 1=2 in (2) gives us the matrix (1).
    [Show full text]