Functions of Difference Matrices Are Toeplitz Plus Hankel
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Quantum Information
Quantum Information J. A. Jones Michaelmas Term 2010 Contents 1 Dirac Notation 3 1.1 Hilbert Space . 3 1.2 Dirac notation . 4 1.3 Operators . 5 1.4 Vectors and matrices . 6 1.5 Eigenvalues and eigenvectors . 8 1.6 Hermitian operators . 9 1.7 Commutators . 10 1.8 Unitary operators . 11 1.9 Operator exponentials . 11 1.10 Physical systems . 12 1.11 Time-dependent Hamiltonians . 13 1.12 Global phases . 13 2 Quantum bits and quantum gates 15 2.1 The Bloch sphere . 16 2.2 Density matrices . 16 2.3 Propagators and Pauli matrices . 18 2.4 Quantum logic gates . 18 2.5 Gate notation . 21 2.6 Quantum networks . 21 2.7 Initialization and measurement . 23 2.8 Experimental methods . 24 3 An atom in a laser field 25 3.1 Time-dependent systems . 25 3.2 Sudden jumps . 26 3.3 Oscillating fields . 27 3.4 Time-dependent perturbation theory . 29 3.5 Rabi flopping and Fermi's Golden Rule . 30 3.6 Raman transitions . 32 3.7 Rabi flopping as a quantum gate . 32 3.8 Ramsey fringes . 33 3.9 Measurement and initialisation . 34 1 CONTENTS 2 4 Spins in magnetic fields 35 4.1 The nuclear spin Hamiltonian . 35 4.2 The rotating frame . 36 4.3 On-resonance excitation . 38 4.4 Excitation phases . 38 4.5 Off-resonance excitation . 39 4.6 Practicalities . 40 4.7 The vector model . 40 4.8 Spin echoes . 41 4.9 Measurement and initialisation . 42 5 Photon techniques 43 5.1 Spatial encoding . -
MATH 237 Differential Equations and Computer Methods
Queen’s University Mathematics and Engineering and Mathematics and Statistics MATH 237 Differential Equations and Computer Methods Supplemental Course Notes Serdar Y¨uksel November 19, 2010 This document is a collection of supplemental lecture notes used for Math 237: Differential Equations and Computer Methods. Serdar Y¨uksel Contents 1 Introduction to Differential Equations 7 1.1 Introduction: ................................... ....... 7 1.2 Classification of Differential Equations . ............... 7 1.2.1 OrdinaryDifferentialEquations. .......... 8 1.2.2 PartialDifferentialEquations . .......... 8 1.2.3 Homogeneous Differential Equations . .......... 8 1.2.4 N-thorderDifferentialEquations . ......... 8 1.2.5 LinearDifferentialEquations . ......... 8 1.3 Solutions of Differential equations . .............. 9 1.4 DirectionFields................................. ........ 10 1.5 Fundamental Questions on First-Order Differential Equations............... 10 2 First-Order Ordinary Differential Equations 11 2.1 ExactDifferentialEquations. ........... 11 2.2 MethodofIntegratingFactors. ........... 12 2.3 SeparableDifferentialEquations . ............ 13 2.4 Differential Equations with Homogenous Coefficients . ................ 13 2.5 First-Order Linear Differential Equations . .............. 14 2.6 Applications.................................... ....... 14 3 Higher-Order Ordinary Linear Differential Equations 15 3.1 Higher-OrderDifferentialEquations . ............ 15 3.1.1 LinearIndependence . ...... 16 3.1.2 WronskianofasetofSolutions . ........ 16 3.1.3 Non-HomogeneousProblem -
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. -
The Exponential of a Matrix
5-28-2012 The Exponential of a Matrix The solution to the exponential growth equation dx kt = kx is given by x = c e . dt 0 It is natural to ask whether you can solve a constant coefficient linear system ′ ~x = A~x in a similar way. If a solution to the system is to have the same form as the growth equation solution, it should look like At ~x = e ~x0. The first thing I need to do is to make sense of the matrix exponential eAt. The Taylor series for ez is ∞ n z z e = . n! n=0 X It converges absolutely for all z. It A is an n × n matrix with real entries, define ∞ n n At t A e = . n! n=0 X The powers An make sense, since A is a square matrix. It is possible to show that this series converges for all t and every matrix A. Differentiating the series term-by-term, ∞ ∞ ∞ ∞ n−1 n n−1 n n−1 n−1 m m d At t A t A t A t A At e = n = = A = A = Ae . dt n! (n − 1)! (n − 1)! m! n=0 n=1 n=1 m=0 X X X X At ′ This shows that e solves the differential equation ~x = A~x. The initial condition vector ~x(0) = ~x0 yields the particular solution At ~x = e ~x0. This works, because e0·A = I (by setting t = 0 in the power series). Another familiar property of ordinary exponentials holds for the matrix exponential: If A and B com- mute (that is, AB = BA), then A B A B e e = e + . -
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 + . -
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. -
The Exponential Function for Matrices
The exponential function for matrices Matrix exponentials provide a concise way of describing the solutions to systems of homoge- neous linear differential equations that parallels the use of ordinary exponentials to solve simple differential equations of the form y0 = λ y. For square matrices the exponential function can be defined by the same sort of infinite series used in calculus courses, but some work is needed in order to justify the construction of such an infinite sum. Therefore we begin with some material needed to prove that certain infinite sums of matrices can be defined in a mathematically sound manner and have reasonable properties. Limits and infinite series of matrices Limits of vector valued sequences in Rn can be defined and manipulated much like limits of scalar valued sequences, the key adjustment being that distances between real numbers that are expressed in the form js−tj are replaced by distances between vectors expressed in the form jx−yj. 1 Similarly, one can talk about convergence of a vector valued infinite series n=0 vn in terms of n the convergence of the sequence of partial sums sn = i=0 vk. As in the case of ordinary infinite series, the best form of convergence is absolute convergence, which correspondsP to the convergence 1 P of the real valued infinite series jvnj with nonnegative terms. A fundamental theorem states n=0 1 that a vector valued infinite series converges if the auxiliary series jvnj does, and there is P n=0 a generalization of the standard M-test: If jvnj ≤ Mn for all n where Mn converges, then P n vn also converges. -
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. -
Approximating the Exponential from a Lie Algebra to a Lie Group
MATHEMATICS OF COMPUTATION Volume 69, Number 232, Pages 1457{1480 S 0025-5718(00)01223-0 Article electronically published on March 15, 2000 APPROXIMATING THE EXPONENTIAL FROM A LIE ALGEBRA TO A LIE GROUP ELENA CELLEDONI AND ARIEH ISERLES 0 Abstract. Consider a differential equation y = A(t; y)y; y(0) = y0 with + y0 2 GandA : R × G ! g,whereg is a Lie algebra of the matricial Lie group G. Every B 2 g canbemappedtoGbythematrixexponentialmap exp (tB)witht 2 R. Most numerical methods for solving ordinary differential equations (ODEs) on Lie groups are based on the idea of representing the approximation yn of + the exact solution y(tn), tn 2 R , by means of exact exponentials of suitable elements of the Lie algebra, applied to the initial value y0. This ensures that yn 2 G. When the exponential is difficult to compute exactly, as is the case when the dimension is large, an approximation of exp (tB) plays an important role in the numerical solution of ODEs on Lie groups. In some cases rational or poly- nomial approximants are unsuitable and we consider alternative techniques, whereby exp (tB) is approximated by a product of simpler exponentials. In this paper we present some ideas based on the use of the Strang splitting for the approximation of matrix exponentials. Several cases of g and G are considered, in tandem with general theory. Order conditions are discussed, and a number of numerical experiments conclude the paper. 1. Introduction Consider the differential equation (1.1) y0 = A(t; y)y; y(0) 2 G; with A : R+ × G ! g; where G is a matricial Lie group and g is the underlying Lie algebra. -
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]. -
Toeplitz and Toeplitz-Block-Toeplitz Matrices and Their Correlation with Syzygies of Polynomials
TOEPLITZ AND TOEPLITZ-BLOCK-TOEPLITZ MATRICES AND THEIR CORRELATION WITH SYZYGIES OF POLYNOMIALS HOUSSAM KHALIL∗, BERNARD MOURRAIN† , AND MICHELLE SCHATZMAN‡ Abstract. In this paper, we re-investigate the resolution of Toeplitz systems T u = g, from a new point of view, by correlating the solution of such problems with syzygies of polynomials or moving lines. We show an explicit connection between the generators of a Toeplitz matrix and the generators of the corresponding module of syzygies. We show that this module is generated by two elements of degree n and the solution of T u = g can be reinterpreted as the remainder of an explicit vector depending on g, by these two generators. This approach extends naturally to multivariate problems and we describe for Toeplitz-block-Toeplitz matrices, the structure of the corresponding generators. Key words. Toeplitz matrix, rational interpolation, syzygie 1. Introduction. Structured matrices appear in various domains, such as scientific computing, signal processing, . They usually express, in a linearize way, a problem which depends on less pa- rameters than the number of entries of the corresponding matrix. An important area of research is devoted to the development of methods for the treatment of such matrices, which depend on the actual parameters involved in these matrices. Among well-known structured matrices, Toeplitz and Hankel structures have been intensively studied [5, 6]. Nearly optimal algorithms are known for the multiplication or the resolution of linear systems, for such structure. Namely, if A is a Toeplitz matrix of size n, multiplying it by a vector or solving a linear system with A requires O˜(n) arithmetic operations (where O˜(n) = O(n logc(n)) for some c > 0) [2, 12]. -
A Note on Multilevel Toeplitz Matrices
Spec. Matrices 2019; 7:114–126 Research Article Open Access Lei Cao and Selcuk Koyuncu* A note on multilevel Toeplitz matrices https://doi.org/110.1515/spma-2019-0011 Received August 7, 2019; accepted September 12, 2019 Abstract: Chien, Liu, Nakazato and Tam proved that all n × n classical Toeplitz matrices (one-level Toeplitz matrices) are unitarily similar to complex symmetric matrices via two types of unitary matrices and the type of the unitary matrices only depends on the parity of n. In this paper we extend their result to multilevel Toeplitz matrices that any multilevel Toeplitz matrix is unitarily similar to a complex symmetric matrix. We provide a method to construct the unitary matrices that uniformly turn any multilevel Toeplitz matrix to a complex symmetric matrix by taking tensor products of these two types of unitary matrices for one-level Toeplitz matrices according to the parity of each level of the multilevel Toeplitz matrices. In addition, we introduce a class of complex symmetric matrices that are unitarily similar to some p-level Toeplitz matrices. Keywords: Multilevel Toeplitz matrix; Unitary similarity; Complex symmetric matrices 1 Introduction Although every complex square matrix is similar to a complex symmetric matrix (see Theorem 4.4.24, [5]), it is known that not every n × n matrix is unitarily similar to a complex symmetric matrix when n ≥ 3 (See [4]). Some characterizations of matrices unitarily equivalent to a complex symmetric matrix (UECSM) were given by [1] and [3]. Very recently, a constructive proof that every Toeplitz matrix is unitarily similar to a complex symmetric matrix was given in [2] in which the unitary matrices turning all n × n Toeplitz matrices to complex symmetric matrices was given explicitly.