Toeplitz Matrices Are Unitarily Similar to Symmetric Matrices by Jianzhen
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Periodic Staircase Matrices and Generalized Cluster Structures
PERIODIC STAIRCASE MATRICES AND GENERALIZED CLUSTER STRUCTURES MISHA GEKHTMAN, MICHAEL SHAPIRO, AND ALEK VAINSHTEIN Abstract. As is well-known, cluster transformations in cluster structures of geometric type are often modeled on determinant identities, such as short Pl¨ucker relations, Desnanot–Jacobi identities and their generalizations. We present a construction that plays a similar role in a description of general- ized cluster transformations and discuss its applications to generalized clus- ter structures in GLn compatible with a certain subclass of Belavin–Drinfeld Poisson–Lie brackets, in the Drinfeld double of GLn, and in spaces of periodic difference operators. 1. Introduction Since the discovery of cluster algebras in [4], many important algebraic varieties were shown to support a cluster structure in a sense that the coordinate rings of such variety is isomorphic to a cluster algebra or an upper cluster algebra. Lie theory and representation theory turned out to be a particularly rich source of varieties of this sort including but in no way limited to such examples as Grassmannians [5, 18], double Bruhat cells [1] and strata in flag varieties [15]. In all these examples, cluster transformations that connect distinguished coordinate charts within a ring of regular functions are modeled on three-term relations such as short Pl¨ucker relations, Desnanot–Jacobi identities and their Lie-theoretic generalizations of the kind considered in [3]. This remains true even in the case of exotic cluster structures on GLn considered in [8, 10] where cluster transformations can be obtained by applying Desnanot–Jacobi type identities to certain structured matrices of a size far exceeding n. -
Arxiv:Math/0010243V1
FOUR SHORT STORIES ABOUT TOEPLITZ MATRIX CALCULATIONS THOMAS STROHMER∗ Abstract. The stories told in this paper are dealing with the solution of finite, infinite, and biinfinite Toeplitz-type systems. A crucial role plays the off-diagonal decay behavior of Toeplitz matrices and their inverses. Classical results of Gelfand et al. on commutative Banach algebras yield a general characterization of this decay behavior. We then derive estimates for the approximate solution of (bi)infinite Toeplitz systems by the finite section method, showing that the approximation rate depends only on the decay of the entries of the Toeplitz matrix and its condition number. Furthermore, we give error estimates for the solution of doubly infinite convolution systems by finite circulant systems. Finally, some quantitative results on the construction of preconditioners via circulant embedding are derived, which allow to provide a theoretical explanation for numerical observations made by some researchers in connection with deconvolution problems. Key words. Toeplitz matrix, Laurent operator, decay of inverse matrix, preconditioner, circu- lant matrix, finite section method. AMS subject classifications. 65T10, 42A10, 65D10, 65F10 0. Introduction. Toeplitz-type equations arise in many applications in mathe- matics, signal processing, communications engineering, and statistics. The excellent surveys [4, 17] describe a number of applications and contain a vast list of references. The stories told in this paper are dealing with the (approximate) solution of biinfinite, infinite, and finite hermitian positive definite Toeplitz-type systems. We pay special attention to Toeplitz-type systems with certain decay properties in the sense that the entries of the matrix enjoy a certain decay rate off the diagonal. -
THREE STEPS on an OPEN ROAD Gilbert Strang This Note Describes
Inverse Problems and Imaging doi:10.3934/ipi.2013.7.961 Volume 7, No. 3, 2013, 961{966 THREE STEPS ON AN OPEN ROAD Gilbert Strang Massachusetts Institute of Technology Cambridge, MA 02139, USA Abstract. This note describes three recent factorizations of banded invertible infinite matrices 1. If A has a banded inverse : A=BC with block{diagonal factors B and C. 2. Permutations factor into a shift times N < 2w tridiagonal permutations. 3. A = LP U with lower triangular L, permutation P , upper triangular U. We include examples and references and outlines of proofs. This note describes three small steps in the factorization of banded matrices. It is written to encourage others to go further in this direction (and related directions). At some point the problems will become deeper and more difficult, especially for doubly infinite matrices. Our main point is that matrices need not be Toeplitz or block Toeplitz for progress to be possible. An important theory is already established [2, 9, 10, 13-16] for non-Toeplitz \band-dominated operators". The Fredholm index plays a key role, and the second small step below (taken jointly with Marko Lindner) computes that index in the special case of permutation matrices. Recall that banded Toeplitz matrices lead to Laurent polynomials. If the scalars or matrices a−w; : : : ; a0; : : : ; aw lie along the diagonals, the polynomial is A(z) = P k akz and the bandwidth is w. The required index is in this case a winding number of det A(z). Factorization into A+(z)A−(z) is a classical problem solved by Plemelj [12] and Gohberg [6-7]. -
Graph Equivalence Classes for Spectral Projector-Based Graph Fourier Transforms Joya A
1 Graph Equivalence Classes for Spectral Projector-Based Graph Fourier Transforms Joya A. Deri, Member, IEEE, and José M. F. Moura, Fellow, IEEE Abstract—We define and discuss the utility of two equiv- Consider a graph G = G(A) with adjacency matrix alence graph classes over which a spectral projector-based A 2 CN×N with k ≤ N distinct eigenvalues and Jordan graph Fourier transform is equivalent: isomorphic equiv- decomposition A = VJV −1. The associated Jordan alence classes and Jordan equivalence classes. Isomorphic equivalence classes show that the transform is equivalent subspaces of A are Jij, i = 1; : : : k, j = 1; : : : ; gi, up to a permutation on the node labels. Jordan equivalence where gi is the geometric multiplicity of eigenvalue 휆i, classes permit identical transforms over graphs of noniden- or the dimension of the kernel of A − 휆iI. The signal tical topologies and allow a basis-invariant characterization space S can be uniquely decomposed by the Jordan of total variation orderings of the spectral components. subspaces (see [13], [14] and Section II). For a graph Methods to exploit these classes to reduce computation time of the transform as well as limitations are discussed. signal s 2 S, the graph Fourier transform (GFT) of [12] is defined as Index Terms—Jordan decomposition, generalized k gi eigenspaces, directed graphs, graph equivalence classes, M M graph isomorphism, signal processing on graphs, networks F : S! Jij i=1 j=1 s ! (s ;:::; s ;:::; s ;:::; s ) ; (1) b11 b1g1 bk1 bkgk I. INTRODUCTION where sij is the (oblique) projection of s onto the Jordan subspace Jij parallel to SnJij. -
Arxiv:2004.05118V1 [Math.RT]
GENERALIZED CLUSTER STRUCTURES RELATED TO THE DRINFELD DOUBLE OF GLn MISHA GEKHTMAN, MICHAEL SHAPIRO, AND ALEK VAINSHTEIN Abstract. We prove that the regular generalized cluster structure on the Drinfeld double of GLn constructed in [9] is complete and compatible with the standard Poisson–Lie structure on the double. Moreover, we show that for n = 4 this structure is distinct from a previously known regular generalized cluster structure on the Drinfeld double, even though they have the same compatible Poisson structure and the same collection of frozen variables. Further, we prove that the regular generalized cluster structure on band periodic matrices constructed in [9] possesses similar compatibility and completeness properties. 1. Introduction In [6, 8] we constructed an initial seed Σn for a complete generalized cluster D structure GCn in the ring of regular functions on the Drinfeld double D(GLn) and proved that this structure is compatible with the standard Poisson–Lie structure on D(GLn). In [9, Section 4] we constructed a different seed Σ¯ n for a regular D D generalized cluster structure GCn on D(GLn). In this note we prove that GCn D shares all the properties of GCn : it is complete and compatible with the standard Poisson–Lie structure on D(GLn). Moreover, we prove that the seeds Σ¯ 4(X, Y ) and T T Σ4(Y ,X ) are not mutationally equivalent. In this way we provide an explicit example of two different regular complete generalized cluster structures on the same variety with the same compatible Poisson structure and the same collection of frozen D variables. -
Polynomials and Hankel Matrices
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Polynomials and Hankel Matrices Miroslav Fiedler Czechoslovak Academy of Sciences Institute of Mathematics iitnci 25 115 67 Praha 1, Czechoslovakia Submitted by V. Ptak ABSTRACT Compatibility of a Hankel n X n matrix W and a polynomial f of degree m, m < n, is defined. If m = n, compatibility means that HC’ = CfH where Cf is the companion matrix of f With a suitable generalization of Cr, this theorem is gener- alized to the case that m < n. INTRODUCTION By a Hankel matrix [S] we shall mean a square complex matrix which has, if of order n, the form ( ai +k), i, k = 0,. , n - 1. If H = (~y~+~) is a singular n X n Hankel matrix, the H-polynomial (Pi of H was defined 131 as the greatest common divisor of the determinants of all (r + 1) x (r + 1) submatrices~of the matrix where r is the rank of H. In other words, (Pi is that polynomial for which the nX(n+l)matrix I, 0 0 0 %fb) 0 i 0 0 0 1 LINEAR ALGEBRA AND ITS APPLICATIONS 66:235-248(1985) 235 ‘F’Elsevier Science Publishing Co., Inc., 1985 52 Vanderbilt Ave., New York, NY 10017 0024.3795/85/$3.30 236 MIROSLAV FIEDLER is the Smith normal form [6] of H,. It has also been shown [3] that qN is a (nonzero) polynomial of degree at most T. It is known [4] that to a nonsingular n X n Hankel matrix H = ((Y~+~)a linear pencil of polynomials of degree at most n can be assigned as follows: f(x) = fo + f,x + . -
The Spectra of Large Toeplitz Band Matrices with A
MATHEMATICS OF COMPUTATION Volume 72, Number 243, Pages 1329{1348 S 0025-5718(03)01505-9 Article electronically published on February 3, 2003 THESPECTRAOFLARGETOEPLITZBANDMATRICES WITH A RANDOMLY PERTURBED ENTRY A. BOTTCHER,¨ M. EMBREE, AND V. I. SOKOLOV Abstract. (j;k) This paper is concerned with the union spΩ Tn(a) of all possible spectra that may emerge when perturbing a large n n Toeplitz band matrix × Tn(a)inthe(j; k) site by a number randomly chosen from some set Ω. The main results give descriptive bounds and, in several interesting situations, even (j;k) provide complete identifications of the limit of sp Tn(a)asn .Also Ω !1 discussed are the cases of small and large sets Ω as well as the \discontinuity of (j;k) the infinite volume case", which means that in general spΩ Tn(a)doesnot converge to something close to sp(j;k) T (a)asn ,whereT (a)isthecor- Ω !1 responding infinite Toeplitz matrix. Illustrations are provided for tridiagonal Toeplitz matrices, a notable special case. 1. Introduction and main results For a complex-valued continuous function a on the complex unit circle T,the infinite Toeplitz matrix T (a) and the finite Toeplitz matrices Tn(a) are defined by n T (a)=(aj k)j;k1 =1 and Tn(a)=(aj k)j;k=1; − − where a` is the `th Fourier coefficient of a, 2π 1 iθ i`θ a` = a(e )e− dθ; ` Z: 2π 2 Z0 Here, we restrict our attention to the case where a is a trigonometric polynomial, a , implying that at most a finite number of the Fourier coefficients are nonzero; equivalently,2P T (a) is a banded matrix. -
Inverse Problems for Hankel and Toeplitz Matrices
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Inverse Problems for Hankel and Toeplitz Matrices Georg Heinig Vniversitiit Leipzig Sektion Mathematik Leipzig 7010, Germany Submitted by Peter Lancaster ABSTRACT The problem is considered how to construct a Toeplitz or Hankel matrix A from one or two equations of the form Au = g. The general solution is described explicitly. Special cases are inverse spectral problems for Hankel and Toeplitz matrices and problems from the theory of orthogonal polynomials. INTRODUCTION When we speak of inverse problems we have in mind the following type of problems: Given vectors uj E C”, gj E C”’ (j = l,.. .,T), we ask for an m x n matrix A of a certain matrix class such that Auj = gj (j=l,...,?-). (0.1) In the present paper we deal with inverse problems with the additional condition that A is a Hankel matrix [ si +j] or a Toeplitz matrix [ ci _j]. Inverse problems for Hankel and Toeplitz matices occur, for example, in the theory of orthogonal polynomials when a measure p on the real line or the unit circle is wanted such that given polynomials are orthogonal with respect to this measure. The moment matrix of p is just the solution of a certain inverse problem and is Hankel (in the real line case) or Toeplitz (in the unit circle case); here the gj are unit vectors. LINEAR ALGEBRA AND ITS APPLICATIONS 165:1-23 (1992) 1 0 Elsevier Science Publishing Co., Inc., 1992 655 Avenue of tbe Americas, New York, NY 10010 0024-3795/92/$5.00 2 GEORG HEINIG Inverse problems for Toeplitz matrices were considered for the first time in the paper [lo]of M. -
Polynomial Sequences Generated by Linear Recurrences
Innocent Ndikubwayo Polynomial Sequences Generated by Linear Recurrences: Location and Reality of Zeros Polynomial Sequences Generated by Linear Recurrences: Location and Reality of Zeros Linear Recurrences: Location by Sequences Generated Polynomial Innocent Ndikubwayo ISBN 978-91-7911-462-6 Department of Mathematics Doctoral Thesis in Mathematics at Stockholm University, Sweden 2021 Polynomial Sequences Generated by Linear Recurrences: Location and Reality of Zeros Innocent Ndikubwayo Academic dissertation for the Degree of Doctor of Philosophy in Mathematics at Stockholm University to be publicly defended on Friday 14 May 2021 at 15.00 in sal 14 (Gradängsalen), hus 5, Kräftriket, Roslagsvägen 101 and online via Zoom, public link is available at the department website. Abstract In this thesis, we study the problem of location of the zeros of individual polynomials in sequences of polynomials generated by linear recurrence relations. In paper I, we establish the necessary and sufficient conditions that guarantee hyperbolicity of all the polynomials generated by a three-term recurrence of length 2, whose coefficients are arbitrary real polynomials. These zeros are dense on the real intervals of an explicitly defined real semialgebraic curve. Paper II extends Paper I to three-term recurrences of length greater than 2. We prove that there always exist non- hyperbolic polynomial(s) in the generated sequence. We further show that with at most finitely many known exceptions, all the zeros of all the polynomials generated by the recurrence lie and are dense on an explicitly defined real semialgebraic curve which consists of real intervals and non-real segments. The boundary points of this curve form a subset of zero locus of the discriminant of the characteristic polynomial of the recurrence. -
Conversion of Sparse Matrix to Band Matrix Using an Fpga For
CONVERSION OF SPARSE MATRIX TO BAND MATRIX USING AN FPGA FOR HIGH-PERFORMANCE COMPUTING by Anjani Chaudhary, M.S. A thesis submitted to the Graduate Council of Texas State University in partial fulfillment of the requirements for the degree of Master of Science with a Major in Engineering December 2020 Committee Members: Semih Aslan, Chair Shuying Sun William A Stapleton COPYRIGHT by Anjani Chaudhary 2020 FAIR USE AND AUTHOR’S PERMISSION STATEMENT Fair Use This work is protected by the Copyright Laws of the United States (Public Law 94-553, section 107). Consistent with fair use as defined in the Copyright Laws, brief quotations from this material are allowed with proper acknowledgement. Use of this material for financial gain without the author’s express written permission is not allowed. Duplication Permission As the copyright holder of this work, I, Anjani Chaudhary, authorize duplication of this work, in whole or in part, for educational or scholarly purposes only. ACKNOWLEDGEMENTS Throughout my research and report writing process, I am grateful to receive support and suggestions from the following people. I would like to thank my supervisor and chair, Dr. Semih Aslan, Associate Professor, Ingram School of Engineering, for his time, support, encouragement, and patience, without which I would not have made it. My committee members Dr. Bill Stapleton, Associate Professor, Ingram School of Engineering, and Dr. Shuying Sun, Associate Professor, Department of Mathematics, for their essential guidance throughout the research and my graduate advisor, Dr. Vishu Viswanathan, for providing the opportunity and valuable suggestions which helped me to come one more step closer to my goal. -
Copyrighted Material
JWST600-IND JWST600-Penney September 28, 2015 7:40 Printer Name: Trim: 6.125in × 9.25in INDEX Additive property of determinants, 244 Cofactor expansion, 240 Argument of a complex number, 297 Column space, 76 Column vector, 2 Band matrix, 407 Complementary vector, 319 Band width, 407 Complex eigenvalue, 302 Basis Complex eigenvector, 302 chain, 438 Complex linear transformation, 304 coordinate matrix for a basis, 217 Complex matrices, 301 coordinate transformation, 223 Complex number definition, 104 argument, 297 normalization, 315 conjugate, 302 ordered, 216 Complex vector space, 303 orthonormal, 315 Conjugate of a complex number, 302 point transformation, 223 Consumption matrix, 198 standard Coordinate matrix, 217 M(m,n), 119 Coordinates n, 119 coordinate vector, 216 Rn, 118 coordinate matrix for a basis, 217 COPYRIGHTEDcoordinate MATERIAL vector, 216 Cn, 302 point matrix, 217 Chain basis, 438 point transformation, 223 Characteristic polynomial, 275 Cramer’s rule, 265 Coefficient matrix, 75 Current, 41 Linear Algebra: Ideas and Applications, Fourth Edition. Richard C. Penney. © 2016 John Wiley & Sons, Inc. Published 2016 by John Wiley & Sons, Inc. Companion website: www.wiley.com/go/penney/linearalgebra 487 JWST600-IND JWST600-Penney September 28, 2015 7:40 Printer Name: Trim: 6.125in × 9.25in 488 INDEX Data compression, 349 Haar function, 346 Determinant Hermitian additive property, 244 adjoint, 412 cofactor expansion, 240 symmetric, 414 Cramer’s rule, 265 Hermitian, orthogonal, 413 inverse matrix, 267 Homogeneous system, 80 Laplace -
Explicit Inverse of a Tridiagonal (P, R)–Toeplitz Matrix
Explicit inverse of a tridiagonal (p; r){Toeplitz matrix A.M. Encinas, M.J. Jim´enez Departament de Matemtiques Universitat Politcnica de Catalunya Abstract Tridiagonal matrices appears in many contexts in pure and applied mathematics, so the study of the inverse of these matrices becomes of specific interest. In recent years the invertibility of nonsingular tridiagonal matrices has been quite investigated in different fields, not only from the theoretical point of view (either in the framework of linear algebra or in the ambit of numerical analysis), but also due to applications, for instance in the study of sound propagation problems or certain quantum oscillators. However, explicit inverses are known only in a few cases, in particular when the tridiagonal matrix has constant diagonals or the coefficients of these diagonals are subjected to some restrictions like the tridiagonal p{Toeplitz matrices [7], such that their three diagonals are formed by p{periodic sequences. The recent formulae for the inversion of tridiagonal p{Toeplitz matrices are based, more o less directly, on the solution of second order linear difference equations, although most of them use a cumbersome formulation, that in fact don not take into account the periodicity of the coefficients. This contribution presents the explicit inverse of a tridiagonal matrix (p; r){Toeplitz, which diagonal coefficients are in a more general class of sequences than periodic ones, that we have called quasi{periodic sequences. A tridiagonal matrix A = (aij) of order n + 2 is called (p; r){Toeplitz if there exists m 2 N0 such that n + 2 = mp and ai+p;j+p = raij; i; j = 0;:::; (m − 1)p: Equivalently, A is a (p; r){Toeplitz matrix iff k ai+kp;j+kp = r aij; i; j = 0; : : : ; p; k = 0; : : : ; m − 1: We have developed a technique that reduces any linear second order difference equation with periodic or quasi-periodic coefficients to a difference equation of the same kind but with constant coefficients [3].