Circulant-Matrices
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Arxiv:2105.00793V3 [Math.NA] 14 Jun 2021 Tubal Matrices
Tubal Matrices Liqun Qi∗ and ZiyanLuo† June 15, 2021 Abstract It was shown recently that the f-diagonal tensor in the T-SVD factorization must satisfy some special properties. Such f-diagonal tensors are called s-diagonal tensors. In this paper, we show that such a discussion can be extended to any real invertible linear transformation. We show that two Eckart-Young like theo- rems hold for a third order real tensor, under any doubly real-preserving unitary transformation. The normalized Discrete Fourier Transformation (DFT) matrix, an arbitrary orthogonal matrix, the product of the normalized DFT matrix and an arbitrary orthogonal matrix are examples of doubly real-preserving unitary transformations. We use tubal matrices as a tool for our study. We feel that the tubal matrix language makes this approach more natural. Key words. Tubal matrix, tensor, T-SVD factorization, tubal rank, B-rank, Eckart-Young like theorems AMS subject classifications. 15A69, 15A18 1 Introduction arXiv:2105.00793v3 [math.NA] 14 Jun 2021 Tensor decompositions have wide applications in engineering and data science [11]. The most popular tensor decompositions include CP decomposition and Tucker decompo- sition as well as tensor train decomposition [11, 3, 17]. The tensor-tensor product (t-product) approach, developed by Kilmer, Martin, Bra- man and others [10, 1, 9, 8], is somewhat different. They defined T-product opera- tion such that a third order tensor can be regarded as a linear operator applied on ∗Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, China; ([email protected]). †Department of Mathematics, Beijing Jiaotong University, Beijing 100044, China. -
Problems in Abstract Algebra
STUDENT MATHEMATICAL LIBRARY Volume 82 Problems in Abstract Algebra A. R. Wadsworth 10.1090/stml/082 STUDENT MATHEMATICAL LIBRARY Volume 82 Problems in Abstract Algebra A. R. Wadsworth American Mathematical Society Providence, Rhode Island Editorial Board Satyan L. Devadoss John Stillwell (Chair) Erica Flapan Serge Tabachnikov 2010 Mathematics Subject Classification. Primary 00A07, 12-01, 13-01, 15-01, 20-01. For additional information and updates on this book, visit www.ams.org/bookpages/stml-82 Library of Congress Cataloging-in-Publication Data Names: Wadsworth, Adrian R., 1947– Title: Problems in abstract algebra / A. R. Wadsworth. Description: Providence, Rhode Island: American Mathematical Society, [2017] | Series: Student mathematical library; volume 82 | Includes bibliographical references and index. Identifiers: LCCN 2016057500 | ISBN 9781470435837 (alk. paper) Subjects: LCSH: Algebra, Abstract – Textbooks. | AMS: General – General and miscellaneous specific topics – Problem books. msc | Field theory and polyno- mials – Instructional exposition (textbooks, tutorial papers, etc.). msc | Com- mutative algebra – Instructional exposition (textbooks, tutorial papers, etc.). msc | Linear and multilinear algebra; matrix theory – Instructional exposition (textbooks, tutorial papers, etc.). msc | Group theory and generalizations – Instructional exposition (textbooks, tutorial papers, etc.). msc Classification: LCC QA162 .W33 2017 | DDC 512/.02–dc23 LC record available at https://lccn.loc.gov/2016057500 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. -
Fourier Transform, Convolution Theorem, and Linear Dynamical Systems April 28, 2016
Mathematical Tools for Neuroscience (NEU 314) Princeton University, Spring 2016 Jonathan Pillow Lecture 23: Fourier Transform, Convolution Theorem, and Linear Dynamical Systems April 28, 2016. Discrete Fourier Transform (DFT) We will focus on the discrete Fourier transform, which applies to discretely sampled signals (i.e., vectors). Linear algebra provides a simple way to think about the Fourier transform: it is simply a change of basis, specifically a mapping from the time domain to a representation in terms of a weighted combination of sinusoids of different frequencies. The discrete Fourier transform is therefore equiv- alent to multiplying by an orthogonal (or \unitary", which is the same concept when the entries are complex-valued) matrix1. For a vector of length N, the matrix that performs the DFT (i.e., that maps it to a basis of sinusoids) is an N × N matrix. The k'th row of this matrix is given by exp(−2πikt), for k 2 [0; :::; N − 1] (where we assume indexing starts at 0 instead of 1), and t is a row vector t=0:N-1;. Recall that exp(iθ) = cos(θ) + i sin(θ), so this gives us a compact way to represent the signal with a linear superposition of sines and cosines. The first row of the DFT matrix is all ones (since exp(0) = 1), and so the first element of the DFT corresponds to the sum of the elements of the signal. It is often known as the \DC component". The next row is a complex sinusoid that completes one cycle over the length of the signal, and each subsequent row has a frequency that is an integer multiple of this \fundamental" frequency. -
The Discrete Fourier Transform
Tutorial 2 - Learning about the Discrete Fourier Transform This tutorial will be about the Discrete Fourier Transform basis, or the DFT basis in short. What is a basis? If we google define `basis', we get: \the underlying support or foundation for an idea, argument, or process". In mathematics, a basis is similar. It is an underlying structure of how we look at something. It is similar to a coordinate system, where we can choose to describe a sphere in either the Cartesian system, the cylindrical system, or the spherical system. They will all describe the same thing, but in different ways. And the reason why we have different systems is because doing things in specific basis is easier than in others. For exam- ple, calculating the volume of a sphere is very hard in the Cartesian system, but easy in the spherical system. When working with discrete signals, we can treat each consecutive element of the vector of values as a consecutive measurement. This is the most obvious basis to look at a signal. Where if we have the vector [1, 2, 3, 4], then at time 0 the value was 1, at the next sampling time the value was 2, and so on, giving us a ramp signal. This is called a time series vector. However, there are also other basis for representing discrete signals, and one of the most useful of these is to use the DFT of the original vector, and to express our data not by the individual values of the data, but by the summation of different frequencies of sinusoids, which make up the data. -
Quantum Fourier Transform Revisited
Quantum Fourier Transform Revisited Daan Camps1,∗, Roel Van Beeumen1, Chao Yang1, 1Computational Research Division, Lawrence Berkeley National Laboratory, CA, United States Abstract The fast Fourier transform (FFT) is one of the most successful numerical algorithms of the 20th century and has found numerous applications in many branches of computational science and engineering. The FFT algorithm can be derived from a particular matrix decomposition of the discrete Fourier transform (DFT) matrix. In this paper, we show that the quantum Fourier transform (QFT) can be derived by further decomposing the diagonal factors of the FFT matrix decomposition into products of matrices with Kronecker product structure. We analyze the implication of this Kronecker product structure on the discrete Fourier transform of rank-1 tensors on a classical computer. We also explain why such a structure can take advantage of an important quantum computer feature that enables the QFT algorithm to attain an exponential speedup on a quantum computer over the FFT algorithm on a classical computer. Further, the connection between the matrix decomposition of the DFT matrix and a quantum circuit is made. We also discuss a natural extension of a radix-2 QFT decomposition to a radix-d QFT decomposition. No prior knowledge of quantum computing is required to understand what is presented in this paper. Yet, we believe this paper may help readers to gain some rudimentary understanding of the nature of quantum computing from a matrix computation point of view. 1 Introduction The fast Fourier transform (FFT) [3] is a widely celebrated algorithmic innovation of the 20th century [19]. The algorithm allows us to perform a discrete Fourier transform (DFT) of a vector of size N in (N log N) O operations. -
Circulant Matrix Constructed by the Elements of One of the Signals and a Vector Constructed by the Elements of the Other Signal
Digital Image Processing Filtering in the Frequency Domain (Circulant Matrices and Convolution) Christophoros Nikou [email protected] University of Ioannina - Department of Computer Science and Engineering 2 Toeplitz matrices • Elements with constant value along the main diagonal and sub-diagonals. • For a NxN matrix, its elements are determined by a (2N-1)-length sequence tn | (N 1) n N 1 T(,)m n t mn t0 t 1 t 2 t(N 1) t t t 1 0 1 T tt22 t1 t t t t N 1 2 1 0 NN C. Nikou – Digital Image Processing (E12) 3 Toeplitz matrices (cont.) • Each row (column) is generated by a shift of the previous row (column). − The last element disappears. − A new element appears. T(,)m n t mn t0 t 1 t 2 t(N 1) t t t 1 0 1 T tt22 t1 t t t t N 1 2 1 0 NN C. Nikou – Digital Image Processing (E12) 4 Circulant matrices • Elements with constant value along the main diagonal and sub-diagonals. • For a NxN matrix, its elements are determined by a N-length sequence cn | 01nN C(,)m n c(m n )mod N c0 cNN 1 c 2 c1 c c c 1 01N C c21 c c02 cN cN 1 c c c c NN1 21 0 NN C. Nikou – Digital Image Processing (E12) 5 Circulant matrices (cont.) • Special case of a Toeplitz matrix. • Each row (column) is generated by a circular shift (modulo N) of the previous row (column). C(,)m n c(m n )mod N c0 cNN 1 c 2 c1 c c c 1 01N C c21 c c02 cN cN 1 c c c c NN1 21 0 NN C. -
Pre- and Post-Processing for Optimal Noise Reduction in Cyclic Prefix
Pre- and Post-Processing for Optimal Noise Reduction in Cyclic Prefix Based Channel Equalizers Bojan Vrcelj and P. P. Vaidyanathan Dept. of Electrical Engineering 136-93 California Institute of Technology Pasadena, CA 91125-0001 Abstract— Cyclic prefix based equalizers are widely used for It is preceded (followed) by the optimal precoder (equalizer) high-speed data transmission over frequency selective channels. for the given input and noise statistics. These blocks are real- Their use in conjunction with DFT filterbanks is especially attrac- ized by constant matrix multiplication, so that the overall com- tive, given the low complexity of implementation. Some examples munications system remains of low complexity. include the DFT-based DMT systems. In this paper we consider In the following we first give a brief overview of the cyclic a general cyclic prefix based system for communication and show prefix system with DFT matrices used as the basic ISI can- that the equalization performance can be improved by simple pre- celer. Then, we introduce a way to deal with noise suppres- and post-processing aimed at reducing the noise at the receiver. This processing is done independently of the ISI cancellation per- sion separately and derive the optimal constrained pair pre- formed by the frequency domain equalizer.1 coder/equalizer for this purpose. The constraint is that in the absence of noise the overall system is still ISI-free. The per- formance of the proposed method is evaluated through com- I. INTRODUCTION puter simulations and a significant improvement with respect to There has been considerable interest in applying the eq– the original system without pre- and post-processing is demon- ualization techniques based on cyclic prefix to high speed data strated. -
Spectral Analysis of the Adjacency Matrix of Random Geometric Graphs
Spectral Analysis of the Adjacency Matrix of Random Geometric Graphs Mounia Hamidouche?, Laura Cottatellucciy, Konstantin Avrachenkov ? Departement of Communication Systems, EURECOM, Campus SophiaTech, 06410 Biot, France y Department of Electrical, Electronics, and Communication Engineering, FAU, 51098 Erlangen, Germany Inria, 2004 Route des Lucioles, 06902 Valbonne, France [email protected], [email protected], [email protected]. Abstract—In this article, we analyze the limiting eigen- multivariate statistics of high-dimensional data. In this case, value distribution (LED) of random geometric graphs the coordinates of the nodes can represent the attributes of (RGGs). The RGG is constructed by uniformly distribut- the data. Then, the metric imposed by the RGG depicts the ing n nodes on the d-dimensional torus Td ≡ [0; 1]d and similarity between the data. connecting two nodes if their `p-distance, p 2 [1; 1] is at In this work, the RGG is constructed by considering a most rn. In particular, we study the LED of the adjacency finite set Xn of n nodes, x1; :::; xn; distributed uniformly and matrix of RGGs in the connectivity regime, in which independently on the d-dimensional torus Td ≡ [0; 1]d. We the average vertex degree scales as log (n) or faster, i.e., choose a torus instead of a cube in order to avoid boundary Ω (log(n)). In the connectivity regime and under some effects. Given a geographical distance, rn > 0, we form conditions on the radius rn, we show that the LED of a graph by connecting two nodes xi; xj 2 Xn if their `p- the adjacency matrix of RGGs converges to the LED of distance, p 2 [1; 1] is at most rn, i.e., kxi − xjkp ≤ rn, the adjacency matrix of a deterministic geometric graph where k:kp is the `p-metric defined as (DGG) with nodes in a grid as n goes to infinity. -
Circulant and Toeplitz Matrices in Compressed Sensing
Circulant and Toeplitz Matrices in Compressed Sensing Holger Rauhut Hausdorff Center for Mathematics & Institute for Numerical Simulation University of Bonn Conference on Time-Frequency Strobl, June 15, 2009 Institute for Numerical Simulation Rheinische Friedrich-Wilhelms-Universität Bonn Holger Rauhut Hausdorff Center for Mathematics & Institute for NumericalCirculant and Simulation Toeplitz University Matrices inof CompressedBonn Sensing Overview • Compressive Sensing (compressive sampling) • Partial Random Circulant and Toeplitz Matrices • Recovery result for `1-minimization • Numerical experiments • Proof sketch (Non-commutative Khintchine inequality) Holger Rauhut Circulant and Toeplitz Matrices 2 Recovery requires the solution of the underdetermined linear system Ax = y: Idea: Sparsity allows recovery of the correct x. Suitable matrices A allow the use of efficient algorithms. Compressive Sensing N Recover a large vector x 2 C from a small number of linear n×N measurements y = Ax with A 2 C a suitable measurement matrix. Usual assumption x is sparse, that is, kxk0 := j supp xj ≤ k where supp x = fj; xj 6= 0g. Interesting case k < n << N. Holger Rauhut Circulant and Toeplitz Matrices 3 Compressive Sensing N Recover a large vector x 2 C from a small number of linear n×N measurements y = Ax with A 2 C a suitable measurement matrix. Usual assumption x is sparse, that is, kxk0 := j supp xj ≤ k where supp x = fj; xj 6= 0g. Interesting case k < n << N. Recovery requires the solution of the underdetermined linear system Ax = y: Idea: Sparsity allows recovery of the correct x. Suitable matrices A allow the use of efficient algorithms. Holger Rauhut Circulant and Toeplitz Matrices 3 n×N For suitable matrices A 2 C , `0 recovers every k-sparse x provided n ≥ 2k. -
1 1.1. the DFT Matrix
FFT January 20, 2016 1 1.1. The DFT matrix. The DFT matrix. By definition, the sequence f(τ)(τ = 0; 1; 2;:::;N − 1), posesses a discrete Fourier transform F (ν)(ν = 0; 1; 2;:::;N − 1), given by − 1 NX1 F (ν) = f(τ)e−i2π(ν=N)τ : (1.1) N τ=0 Of course, this definition can be immediately rewritten in the matrix form as follows 2 3 2 3 F (1) f(1) 6 7 6 7 6 7 6 7 6 F (2) 7 1 6 f(2) 7 6 . 7 = p F 6 . 7 ; (1.2) 4 . 5 N 4 . 5 F (N − 1) f(N − 1) where the DFT (i.e., the discrete Fourier transform) matrix is defined by 2 3 1 1 1 1 · 1 6 − 7 6 1 w w2 w3 ··· wN 1 7 6 7 h i 6 1 w2 w4 w6 ··· w2(N−1) 7 1 − − 1 6 7 F = p w(k 1)(j 1) = p 6 3 6 9 ··· 3(N−1) 7 N 1≤k;j≤N N 6 1 w w w w 7 6 . 7 4 . 5 1 wN−1 w2(N−1) w3(N−1) ··· w(N−1)(N−1) (1.3) 2πi with w = e N being the primitive N-th root of unity. 1.2. The IDFT matrix. To recover N values of the function from its discrete Fourier transform we simply have to invert the DFT matrix to obtain 2 3 2 3 f(1) F (1) 6 7 6 7 6 f(2) 7 p 6 F (2) 7 6 7 −1 6 7 6 . -
Analytical Solution of the Symmetric Circulant Tridiagonal Linear System
Rose-Hulman Institute of Technology Rose-Hulman Scholar Mathematical Sciences Technical Reports (MSTR) Mathematics 8-24-2014 Analytical Solution of the Symmetric Circulant Tridiagonal Linear System Sean A. Broughton Rose-Hulman Institute of Technology, [email protected] Jeffery J. Leader Rose-Hulman Institute of Technology, [email protected] Follow this and additional works at: https://scholar.rose-hulman.edu/math_mstr Part of the Numerical Analysis and Computation Commons Recommended Citation Broughton, Sean A. and Leader, Jeffery J., "Analytical Solution of the Symmetric Circulant Tridiagonal Linear System" (2014). Mathematical Sciences Technical Reports (MSTR). 103. https://scholar.rose-hulman.edu/math_mstr/103 This Article is brought to you for free and open access by the Mathematics at Rose-Hulman Scholar. It has been accepted for inclusion in Mathematical Sciences Technical Reports (MSTR) by an authorized administrator of Rose-Hulman Scholar. For more information, please contact [email protected]. Analytical Solution of the Symmetric Circulant Tridiagonal Linear System S. Allen Broughton and Jeffery J. Leader Mathematical Sciences Technical Report Series MSTR 14-02 August 24, 2014 Department of Mathematics Rose-Hulman Institute of Technology http://www.rose-hulman.edu/math.aspx Fax (812)-877-8333 Phone (812)-877-8193 Analytical Solution of the Symmetric Circulant Tridiagonal Linear System S. Allen Broughton Rose-Hulman Institute of Technology Jeffery J. Leader Rose-Hulman Institute of Technology August 24, 2014 Abstract A circulant tridiagonal system is a special type of Toeplitz system that appears in a variety of problems in scientific computation. In this paper we give a formula for the inverse of a symmetric circulant tridiagonal matrix as a product of a circulant matrix and its transpose, and discuss the utility of this approach for solving the associated system. -
Discovering Transforms: a Tutorial on Circulant Matrices, Circular Convolution, and the Discrete Fourier Transform
DISCOVERING TRANSFORMS: A TUTORIAL ON CIRCULANT MATRICES, CIRCULAR CONVOLUTION, AND THE DISCRETE FOURIER TRANSFORM BASSAM BAMIEH∗ Key words. Discrete Fourier Transform, Circulant Matrix, Circular Convolution, Simultaneous Diagonalization of Matrices, Group Representations AMS subject classifications. 42-01,15-01, 42A85, 15A18, 15A27 Abstract. How could the Fourier and other transforms be naturally discovered if one didn't know how to postulate them? In the case of the Discrete Fourier Transform (DFT), we show how it arises naturally out of analysis of circulant matrices. In particular, the DFT can be derived as the change of basis that simultaneously diagonalizes all circulant matrices. In this way, the DFT arises naturally from a linear algebra question about a set of matrices. Rather than thinking of the DFT as a signal transform, it is more natural to think of it as a single change of basis that renders an entire set of mutually-commuting matrices into simple, diagonal forms. The DFT can then be \discovered" by solving the eigenvalue/eigenvector problem for a special element in that set. A brief outline is given of how this line of thinking can be generalized to families of linear operators, leading to the discovery of the other common Fourier-type transforms, as well as its connections with group representations theory. 1. Introduction. The Fourier transform in all its forms is ubiquitous. Its many useful properties are introduced early on in Mathematics, Science and Engineering curricula [1]. Typically, it is introduced as a transformation on functions or signals, and then its many useful properties are easily derived. Those properties are then shown to be remarkably effective in solving certain differential equations, or in an- alyzing the action of time-invariant linear dynamical systems, amongst many other uses.