Discovering Transforms: a Tutorial on Circulant Matrices, Circular Convolution, and the Discrete Fourier Transform
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Simultaneous Diagonalization and SVD of Commuting Matrices
Simultaneous Diagonalization and SVD of Commuting Matrices Ronald P. Nordgren1 Brown School of Engineering, Rice University Abstract. We present a matrix version of a known method of constructing common eigenvectors of two diagonalizable commuting matrices, thus enabling their simultane- ous diagonalization. The matrices may have simple eigenvalues of multiplicity greater than one. The singular value decomposition (SVD) of a class of commuting matrices also is treated. The effect of row/column permutation is examined. Examples are given. 1 Introduction It is well known that if two diagonalizable matrices have the same eigenvectors, then they commute. The converse also is true and a construction for the common eigen- vectors (enabling simultaneous diagonalization) is known. If one of the matrices has distinct eigenvalues (multiplicity one), it is easy to show that its eigenvectors pertain to both commuting matrices. The case of matrices with simple eigenvalues of mul- tiplicity greater than one requires a more complicated construction of their common eigenvectors. Here, we present a matrix version of a construction procedure given by Horn and Johnson [1, Theorem 1.3.12] and in a video by Sadun [4]. The eigenvector construction procedure also is applied to the singular value decomposition of a class of commuting matrices that includes the case where at least one of the matrices is real and symmetric. In addition, we consider row/column permutation of the commuting matrices. Three examples illustrate the eigenvector construction procedure. 2 Eigenvector Construction Let A and B be diagonalizable square matrices that commute, i.e. AB = BA (1) and their Jordan canonical forms read −1 −1 A = SADASA , B = SBDB SB . -
The Algebra Generated by Three Commuting Matrices
THE ALGEBRA GENERATED BY THREE COMMUTING MATRICES B.A. SETHURAMAN Abstract. We present a survey of an open problem concerning the dimension of the algebra generated by three commuting matrices. This article concerns a problem in algebra that is completely elementary to state, yet, has proven tantalizingly difficult and is as yet unsolved. Consider C[A; B; C] , the C-subalgebra of the n × n matrices Mn(C) generated by three commuting matrices A, B, and C. Thus, C[A; B; C] consists of all C- linear combinations of \monomials" AiBjCk, where i, j, and k range from 0 to infinity. Note that C[A; B; C] and Mn(C) are naturally vector-spaces over C; moreover, C[A; B; C] is a subspace of Mn(C). The problem, quite simply, is this: Is the dimension of C[A; B; C] as a C vector space bounded above by n? 2 Note that the dimension of C[A; B; C] is at most n , because the dimen- 2 sion of Mn(C) is n . Asking for the dimension of C[A; B; C] to be bounded above by n when A, B, and C commute is to put considerable restrictions on C[A; B; C]: this is to require that C[A; B; C] occupy only a small portion of the ambient Mn(C) space in which it sits. Actually, the dimension of C[A; B; C] is already bounded above by some- thing slightly smaller than n2, thanks to a classical theorem of Schur ([16]), who showed that the maximum possible dimension of a commutative C- 2 subalgebra of Mn(C) is 1 + bn =4c. -
Arxiv:1705.10957V2 [Math.AC] 14 Dec 2017 Ideal Called Is and Commutator X Y Vrafield a Over Let Matrix Eae Leri Es Is E Sdfierlvn Notions
NEARLY COMMUTING MATRICES ZHIBEK KADYRSIZOVA Abstract. We prove that the algebraic set of pairs of matrices with a diag- onal commutator over a field of positive prime characteristic, its irreducible components, and their intersection are F -pure when the size of matrices is equal to 3. Furthermore, we show that this algebraic set is reduced and the intersection of its irreducible components is irreducible in any characteristic for pairs of matrices of any size. In addition, we discuss various conjectures on the singularities of these algebraic sets and the system of parameters on the corresponding coordinate rings. Keywords: Frobenius, singularities, F -purity, commuting matrices 1. Introduction and preliminaries In this paper we study algebraic sets of pairs of matrices such that their commutator is either nonzero diagonal or zero. We also consider some other related algebraic sets. First let us define relevant notions. Let X =(x ) and Y =(y ) be n×n matrices of indeterminates arXiv:1705.10957v2 [math.AC] 14 Dec 2017 ij 1≤i,j≤n ij 1≤i,j≤n over a field K. Let R = K[X,Y ] be the polynomial ring in {xij,yij}1≤i,j≤n and let I denote the ideal generated by the off-diagonal entries of the commutator matrix XY − YX and J denote the ideal generated by the entries of XY − YX. The ideal I defines the algebraic set of pairs of matrices with a diagonal commutator and is called the algebraic set of nearly commuting matrices. The ideal J defines the algebraic set of pairs of commuting matrices. -
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. -
Lecture 2: Spectral Theorems
Lecture 2: Spectral Theorems This lecture introduces normal matrices. The spectral theorem will inform us that normal matrices are exactly the unitarily diagonalizable matrices. As a consequence, we will deduce the classical spectral theorem for Hermitian matrices. The case of commuting families of matrices will also be studied. All of this corresponds to section 2.5 of the textbook. 1 Normal matrices Definition 1. A matrix A 2 Mn is called a normal matrix if AA∗ = A∗A: Observation: The set of normal matrices includes all the Hermitian matrices (A∗ = A), the skew-Hermitian matrices (A∗ = −A), and the unitary matrices (AA∗ = A∗A = I). It also " # " # 1 −1 1 1 contains other matrices, e.g. , but not all matrices, e.g. 1 1 0 1 Here is an alternate characterization of normal matrices. Theorem 2. A matrix A 2 Mn is normal iff ∗ n kAxk2 = kA xk2 for all x 2 C : n Proof. If A is normal, then for any x 2 C , 2 ∗ ∗ ∗ ∗ ∗ 2 kAxk2 = hAx; Axi = hx; A Axi = hx; AA xi = hA x; A xi = kA xk2: ∗ n n Conversely, suppose that kAxk = kA xk for all x 2 C . For any x; y 2 C and for λ 2 C with jλj = 1 chosen so that <(λhx; (A∗A − AA∗)yi) = jhx; (A∗A − AA∗)yij, we expand both sides of 2 ∗ 2 kA(λx + y)k2 = kA (λx + y)k2 to obtain 2 2 ∗ 2 ∗ 2 ∗ ∗ kAxk2 + kAyk2 + 2<(λhAx; Ayi) = kA xk2 + kA yk2 + 2<(λhA x; A yi): 2 ∗ 2 2 ∗ 2 Using the facts that kAxk2 = kA xk2 and kAyk2 = kA yk2, we derive 0 = <(λhAx; Ayi − λhA∗x; A∗yi) = <(λhx; A∗Ayi − λhx; AA∗yi) = <(λhx; (A∗A − AA∗)yi) = jhx; (A∗A − AA∗)yij: n ∗ ∗ n Since this is true for any x 2 C , we deduce (A A − AA )y = 0, which holds for any y 2 C , meaning that A∗A − AA∗ = 0, as desired. -
On Low Rank Perturbations of Complex Matrices and Some Discrete Metric Spaces∗
Electronic Journal of Linear Algebra ISSN 1081-3810 A publication of the International Linear Algebra Society Volume 18, pp. 302-316, June 2009 ELA ON LOW RANK PERTURBATIONS OF COMPLEX MATRICES AND SOME DISCRETE METRIC SPACES∗ LEV GLEBSKY† AND LUIS MANUEL RIVERA‡ Abstract. In this article, several theorems on perturbations ofa complex matrix by a matrix ofa given rank are presented. These theorems may be divided into two groups. The first group is about spectral properties ofa matrix under such perturbations; the second is about almost-near relations with respect to the rank distance. Key words. Complex matrices, Rank, Perturbations. AMS subject classifications. 15A03, 15A18. 1. Introduction. In this article, we present several theorems on perturbations of a complex matrix by a matrix of a given rank. These theorems may be divided into two groups. The first group is about spectral properties of a matrix under such perturbations; the second is about almost-near relations with respect to the rank distance. 1.1. The first group. Theorem 2.1gives necessary and sufficient conditions on the Weyr characteristics, [20], of matrices A and B if rank(A − B) ≤ k.In one direction the theorem is known; see [12, 13, 14]. For k =1thetheoremisa reformulation of a theorem of Thompson [16] (see Theorem 2.3 of the present article). We prove Theorem 2.1by induction with respect to k. To this end, we introduce a discrete metric space (the space of Weyr characteristics) and prove that this metric space is geodesic. In fact, the induction (with respect to k) is hidden in this proof (see Proposition 3.6 and Proposition 3.1). -
Representation Theory
M392C NOTES: REPRESENTATION THEORY ARUN DEBRAY MAY 14, 2017 These notes were taken in UT Austin's M392C (Representation Theory) class in Spring 2017, taught by Sam Gunningham. I live-TEXed them using vim, so there may be typos; please send questions, comments, complaints, and corrections to [email protected]. Thanks to Kartik Chitturi, Adrian Clough, Tom Gannon, Nathan Guermond, Sam Gunningham, Jay Hathaway, and Surya Raghavendran for correcting a few errors. Contents 1. Lie groups and smooth actions: 1/18/172 2. Representation theory of compact groups: 1/20/174 3. Operations on representations: 1/23/176 4. Complete reducibility: 1/25/178 5. Some examples: 1/27/17 10 6. Matrix coefficients and characters: 1/30/17 12 7. The Peter-Weyl theorem: 2/1/17 13 8. Character tables: 2/3/17 15 9. The character theory of SU(2): 2/6/17 17 10. Representation theory of Lie groups: 2/8/17 19 11. Lie algebras: 2/10/17 20 12. The adjoint representations: 2/13/17 22 13. Representations of Lie algebras: 2/15/17 24 14. The representation theory of sl2(C): 2/17/17 25 15. Solvable and nilpotent Lie algebras: 2/20/17 27 16. Semisimple Lie algebras: 2/22/17 29 17. Invariant bilinear forms on Lie algebras: 2/24/17 31 18. Classical Lie groups and Lie algebras: 2/27/17 32 19. Roots and root spaces: 3/1/17 34 20. Properties of roots: 3/3/17 36 21. Root systems: 3/6/17 37 22. Dynkin diagrams: 3/8/17 39 23. -
Lecture Notes: Basic Group and Representation Theory
Lecture notes: Basic group and representation theory Thomas Willwacher February 27, 2014 2 Contents 1 Introduction 5 1.1 Definitions . .6 1.2 Actions and the orbit-stabilizer Theorem . .8 1.3 Generators and relations . .9 1.4 Representations . 10 1.5 Basic properties of representations, irreducibility and complete reducibility . 11 1.6 Schur’s Lemmata . 12 2 Finite groups and finite dimensional representations 15 2.1 Character theory . 15 2.2 Algebras . 17 2.3 Existence and classification of irreducible representations . 18 2.4 How to determine the character table – Burnside’s algorithm . 20 2.5 Real and complex representations . 21 2.6 Induction, restriction and characters . 23 2.7 Exercises . 25 3 Representation theory of the symmetric groups 27 3.1 Notations . 27 3.2 Conjugacy classes . 28 3.3 Irreducible representations . 29 3.4 The Frobenius character formula . 31 3.5 The hook lengths formula . 34 3.6 Induction and restriction . 34 3.7 Schur-Weyl duality . 35 4 Lie groups, Lie algebras and their representations 37 4.1 Overview . 37 4.2 General definitions and facts about Lie algebras . 39 4.3 The theorems of Lie and Engel . 40 4.4 The Killing form and Cartan’s criteria . 41 4.5 Classification of complex simple Lie algebras . 43 4.6 Classification of real simple Lie algebras . 44 4.7 Generalities on representations of Lie algebras . 44 4.8 Representation theory of sl(2; C) ................................ 45 4.9 General structure theory of semi-simple Lie algebras . 48 4.10 Representation theory of complex semi-simple Lie algebras . -
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. -
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. -
Linear Algebra: Example Sheet 3 of 4
Michaelmas Term 2020 Linear Algebra: Example Sheet 3 of 4 1. Find the eigenvalues and give bases for the eigenspaces of the following complex matrices: 0 1 1 0 1 0 1 1 −1 1 0 1 1 −1 1 @ 0 3 −2 A ; @ 0 3 −2 A ; @ −1 3 −1 A : 0 1 0 0 1 0 −1 1 1 The second and third matrices commute; find a basis with respect to which they are both diagonal. 2. By considering the rank or minimal polynomial of a suitable matrix, find the eigenvalues of the n × n matrix A with each diagonal entry equal to λ and all other entries 1. Hence write down the determinant of A. 3. Let A be an n × n matrix all the entries of which are real. Show that the minimum polynomial of A, over the complex numbers, has real coefficients. 4. (i) Let V be a vector space, let π1; π2; : : : ; πk be endomorphisms of V such that idV = π1 + ··· + πk and πiπj = 0 for any i 6= j. Show that V = U1 ⊕ · · · ⊕ Uk, where Uj = Im(πj). (ii) Let α be an endomorphism of V satisfying the equation α3 = α. By finding suitable endomorphisms of V depending on α, use (i) to prove that V = V0 ⊕ V1 ⊕ V−1, where Vλ is the λ-eigenspace of α. (iii) Apply (i) to prove the Generalised Eigenspace Decomposition Theorem stated in lectures. [Hint: It may help to re-read the proof of the Diagonalisability Theorem given in lectures. You may use the fact that if p1(t); : : : ; pk(t) 2 C[t] are non-zero polynomials with no common factor, Euclid's algorithm implies that there exist polynomials q1(t); : : : ; qk(t) 2 C[t] such that p1(t)q1(t) + ::: + pk(t)qk(t) = 1.] 5. -
SIMILARITY OP SETS of MATRICES OVER a SKEW FIELD Ph. D. Thesis WALTER SCOTT SIZER Bedford College, University of London
SIMILARITY OP SETS OF MATRICES OVER A SKEW FIELD Ph. D. Thesis WALTER SCOTT SIZER Bedford College, University of London ProQuest Number: 10098309 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. uest. ProQuest 10098309 Published by ProQuest LLC(2016). Copyright of the Dissertation is held by the Author. All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code. Microform Edition © ProQuest LLC. ProQuest LLC 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 48106-1346 - 1- ABSTRAGT This thesis looks at various questions in matrix theory over skew, fields. The common thread in all these considera tions is the determination of an easily described form for a set of matrices, as simultaneously upper triangular or diagonal, for example. The first chapter, in addition to giving some results which prove useful in later chapters, describes the work of P. M. Cohn on the normal form of a single matrix over a skew field. We use these results to show that, if the skew field D has a perfect center, then any matrix over D is similar to a matrix with entries in a commutative field. The second chapter gives some results concerning com mutativity, including the upper triangularizability of any set of commuting matrices, conditions allowing the simul taneous diagonalization of a set of commuting diagonal- izable matrices, and a description, over skew fields with perfect centers, of matrices commuting with a given matrix.