DOCSLIB.ORG

Linear Algebra Review

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Linear Algebra Review Linear Programming Lecture 1: Linear Algebra Review Lecture 1: Linear Algebra Review Linear Programming 1 / 24 1 Linear Algebra Review 2 Linear Algebra Review 3 Block Structured Matrices 4 Gaussian Elimination Matrices 5 Gauss-Jordan Elimination (Pivoting) Lecture 1: Linear Algebra Review Linear Programming 2 / 24 Matrices in Rm×n A 2 Rm×n columns rows 2 3 2 3 a11 a12 ::: a1n a1• 6 a21 a22 ::: a2n 7 6 a2• 7 A = 6 7 = a a ::: a = 6 7 6 . .. 7 •1 •2 •n 6 . 7 4 . 5 4 . 5 am1 am2 ::: amn am• 2 3 2 T 3 a11 a21 ::: am1 a•1 T 6 a12 a22 ::: am2 7 6 a•2 7 AT = 6 7 = 6 7 = aT aT ::: aT 6 . .. 7 6 . 7 1• 2• m• 4 . 5 4 . 5 T a1n a2n ::: amn a•n Lecture 1: Linear Algebra Review Linear Programming 3 / 24 Matrix Vector Multiplication A column space view of matrix vector multiplication. 2 a11 a12 ::: a1n 3 2 x1 3 2 a11 3 2 a12 3 2 a1n 3 6 a21 a22 ::: a2n 7 6 x2 7 6 a21 7 6 a22 7 6 a2n 7 6 7 6 7 = x 6 7+ x 6 7+ ··· + x 6 7 6 . .. 7 6 . 7 1 6 . 7 2 6 . 7 n 6 . 7 4 . 5 4 . 5 4 . 5 4 . 5 4 . 5 am1 am2 ::: amn xn am1 am2 amn = x1 a•1 + x2 a•2 + ··· + xn a•n A linear combination of the columns. Lecture 1: Linear Algebra Review Linear Programming 4 / 24 The Range of a Matrix Let A 2 Rm×n (an m × n matrix having real entries). Range of A m n Ran (A) = fy 2 R j 9 x 2 R such that y = Ax g Ran (A) = the linear span of the columns of A Lecture 1: Linear Algebra Review Linear Programming 5 / 24 Two Special Subspaces n Let v1;:::; vk 2 R . The linear span of v1;:::; vk : Span [v1;:::; vk ] = fy j y = ξ1v1 + ξ2v2 + ··· + ξk vk ; ξ1; : : : ; ξk 2 Rg The subspace orthogonal to v1;:::; vk : ? n fv1;:::; vk g = fz 2 R j z • vi = 0; i = 1;:::; k g ? ? Facts: fv1;:::; vk g = Span [v1;:::; vk ] ? h ?i Span [v1;:::; vk ] = Span [v1;:::; vk ] Lecture 1: Linear Algebra Review Linear Programming 6 / 24 Matrix Vector Multiplication A row space view of matrix vector multiplication. 2 3 2 3 2 3 2 Pn 3 a11 a12 ::: a1n x1 a1• • x i=1 a1i xi Pn 6 a21 a22 ::: a2n 7 6 x2 7 6 a2• • x 7 6 i=1 a2i xi 7 6 7 6 7 = 6 7 = 6 7 6 . .. 7 6 . 7 6 . 7 6 . 7 4 . 5 4 . 5 4 . 5 4 . 5 Pn am1 am2 ::: amn xn am• • x i=1 ami xi The dot product of x with the rows of A. Lecture 1: Linear Algebra Review Linear Programming 7 / 24 Fundamental Theorem of the Alternative: ? ? Nul (A) = Ran AT Ran (A) = Nul AT The Null Space of a Matrix Let A 2 Rm×n (an m × n matrix having real entries). Null Space of A n Nul (A) = fx 2 R j Ax = 0g Nul (A) = subspace orthogonal to the rows of A ? = Span [a1•; a2•;:::; am•] ? = Ran AT Lecture 1: Linear Algebra Review Linear Programming 8 / 24 The Null Space of a Matrix Let A 2 Rm×n (an m × n matrix having real entries). Null Space of A n Nul (A) = fx 2 R j Ax = 0g Nul (A) = subspace orthogonal to the rows of A ? = Span [a1•; a2•;:::; am•] ? = Ran AT Fundamental Theorem of the Alternative: ? ? Nul (A) = Ran AT Ran (A) = Nul AT Lecture 1: Linear Algebra Review Linear Programming 8 / 24 2 3 −4 1 1 0 0 3 6 2 2 0 0 1 0 7 6 7 B I3×3 =6 −1 0 0 0 0 1 7= 6 7 02×3 C 4 0 0 0 2 1 4 5 0 0 0 1 0 3 where 2 3 −4 1 3 2 1 4 B = 2 2 0 ; C = 4 5 1 0 3 −1 0 0 Block Structured Matrices 2 3 −4 1 1 0 0 3 6 2 2 0 0 1 0 7 6 7 A=6 −1 0 0 0 0 1 7 6 7 4 0 0 0 2 1 4 5 0 0 0 1 0 3 Lecture 1: Linear Algebra Review Linear Programming 9 / 24 B I = 3×3 02×3 C where 2 3 −4 1 3 2 1 4 B = 2 2 0 ; C = 4 5 1 0 3 −1 0 0 Block Structured Matrices 2 3 −4 1 1 0 0 3 2 3 −4 1 1 0 0 3 6 2 2 0 0 1 0 7 6 2 2 0 0 1 0 7 6 7 6 7 A=6 −1 0 0 0 0 1 7=6 −1 0 0 0 0 1 7 6 7 6 7 4 0 0 0 2 1 4 5 4 0 0 0 2 1 4 5 0 0 0 1 0 3 0 0 0 1 0 3 Lecture 1: Linear Algebra Review Linear Programming 9 / 24 Block Structured Matrices 2 3 −4 1 1 0 0 3 2 3 −4 1 1 0 0 3 6 2 2 0 0 1 0 7 6 2 2 0 0 1 0 7 6 7 6 7 B I3×3 A=6 −1 0 0 0 0 1 7=6 −1 0 0 0 0 1 7= 6 7 6 7 02×3 C 4 0 0 0 2 1 4 5 4 0 0 0 2 1 4 5 0 0 0 1 0 3 0 0 0 1 0 3 where 2 3 −4 1 3 2 1 4 B = 2 2 0 ; C = 4 5 1 0 3 −1 0 0 Lecture 1: Linear Algebra Review Linear Programming 9 / 24 Can we exploit the structure of A? Multiplication of Block Structured Matrices Consider the matrix product AM, where 2 1 2 3 2 3 −4 1 1 0 0 3 0 4 2 2 0 0 1 0 6 7 6 7 6 −1 −1 7 A = 6 −1 0 0 0 0 1 7 and M = 6 7 6 7 6 2 −1 7 6 0 0 0 2 1 4 7 6 7 4 5 6 4 3 7 0 0 0 1 0 3 4 5 −2 0 Lecture 1: Linear Algebra Review Linear Programming 10 / 24 Multiplication of Block Structured Matrices Consider the matrix product AM, where 2 1 2 3 2 3 −4 1 1 0 0 3 0 4 2 2 0 0 1 0 6 7 6 7 6 −1 −1 7 A = 6 −1 0 0 0 0 1 7 and M = 6 7 6 7 6 2 −1 7 6 0 0 0 2 1 4 7 6 7 4 5 6 4 3 7 0 0 0 1 0 3 4 5 −2 0 Can we exploit the structure of A? Lecture 1: Linear Algebra Review Linear Programming 10 / 24 BI X A = 3×3 so take M = ; 02×3 C Y 2 1 2 3 2 2 −1 3 where X = 4 0 4 5 ; and Y = 4 4 3 5 : −1 −1 −2 0 Multiplication of Block Structured Matrices Consider the matrix product AM, where 2 1 2 3 2 3 −4 1 1 0 0 3 0 4 2 2 0 0 1 0 6 7 6 7 6 −1 −1 7 A = 6 −1 0 0 0 0 1 7 and M = 6 7 6 7 6 2 −1 7 6 0 0 0 2 1 4 7 6 7 4 5 6 4 3 7 0 0 0 1 0 3 4 5 −2 0 Can we exploit the structure of A? Lecture 1: Linear Algebra Review Linear Programming 11 / 24 X so take M = ; Y 2 1 2 3 2 2 −1 3 where X = 4 0 4 5 ; and Y = 4 4 3 5 : −1 −1 −2 0 Multiplication of Block Structured Matrices Consider the matrix product AM, where 2 1 2 3 2 3 −4 1 1 0 0 3 0 4 2 2 0 0 1 0 6 7 6 7 6 −1 −1 7 A = 6 −1 0 0 0 0 1 7 and M = 6 7 6 7 6 2 −1 7 6 0 0 0 2 1 4 7 6 7 4 5 6 4 3 7 0 0 0 1 0 3 4 5 −2 0 Can we exploit the structure of A? BI A = 3×3 02×3 C Lecture 1: Linear Algebra Review Linear Programming 11 / 24 Multiplication of Block Structured Matrices Consider the matrix product AM, where 2 1 2 3 2 3 −4 1 1 0 0 3 0 4 2 2 0 0 1 0 6 7 6 7 6 −1 −1 7 A = 6 −1 0 0 0 0 1 7 and M = 6 7 6 7 6 2 −1 7 6 0 0 0 2 1 4 7 6 7 4 5 6 4 3 7 0 0 0 1 0 3 4 5 −2 0 Can we exploit the structure of A? BI X A = 3×3 so take M = ; 02×3 C Y 2 1 2 3 2 2 −1 3 where X = 4 0 4 5 ; and Y = 4 4 3 5 : −1 −1 −2 0 Lecture 1: Linear Algebra Review Linear Programming 11 / 24 2 2 2 −11 3 2 2 −1 3 3 6 4 2 12 5 + 4 4 3 5 7 6 7 6 −1 −2 −2 0 7 = 6 7 6 7 6 7 4 4 1 5 −4 −1 2 4 −12 3 6 6 15 7 6 7 = 6 1 −2 7 : 6 7 4 4 1 5 −4 −1 Multiplication of Block Structured Matrices BI X BX + Y AM = 3×3 = 02×3 C Y CY Lecture 1: Linear Algebra Review Linear Programming 12 / 24 Multiplication of Block Structured Matrices BI X BX + Y AM = 3×3 = 02×3 C Y CY 2 2 2 −11 3 2 2 −1 3 3 6 4 2 12 5 + 4 4 3 5 7 6 7 6 −1 −2 −2 0 7 = 6 7 6 7 6 7 4 4 1 5 −4 −1 2 4 −12 3 6 6 15 7 6 7 = 6 1 −2 7 : 6 7 4 4 1 5 −4 −1 Lecture 1: Linear Algebra Review Linear Programming 12 / 24 The solution set is either empty, a single point, or an infinite set n If a solution x0 2 R exists, then the set of solutions is given by x0 + Nul (A) : Solving Systems of Linear equations Let A 2 Rm×n and b 2 Rm.
Load more

Recommended publications
  • Quantum Cluster Algebras and Quantum Nilpotent Algebras
    Quantum cluster algebras and quantum nilpotent algebras K. R. Goodearl ∗ and M. T. Yakimov † ∗Department of Mathematics, University of California, Santa Barbara, CA 93106, U.S.A., and †Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, U.S.A. Proceedings of the National Academy of Sciences of the United States of America 111 (2014) 9696–9703 A major direction in the theory of cluster algebras is to construct construction does not rely on any initial combinatorics of the (quantum) cluster algebra structures on the (quantized) coordinate algebras. On the contrary, the construction itself produces rings of various families of varieties arising in Lie theory. We prove intricate combinatorial data for prime elements in chains of that all algebras in a very large axiomatically defined class of noncom- subalgebras. When this is applied to special cases, we re- mutative algebras possess canonical quantum cluster algebra struc- cover the Weyl group combinatorics which played a key role tures. Furthermore, they coincide with the corresponding upper quantum cluster algebras. We also establish analogs of these results in categorification earlier [7, 9, 10]. Because of this, we expect for a large class of Poisson nilpotent algebras. Many important fam- that our construction will be helpful in building a unified cat- ilies of coordinate rings are subsumed in the class we are covering, egorification of quantum nilpotent algebras. Finally, we also which leads to a broad range of application of the general results prove similar results for (commutative) cluster algebras using to the above mentioned types of problems. As a consequence, we Poisson prime elements.
    [Show full text]
  • Introduction to Cluster Algebras. Chapters
    Introduction to Cluster Algebras Chapters 1–3 (preliminary version) Sergey Fomin Lauren Williams Andrei Zelevinsky arXiv:1608.05735v4 [math.CO] 30 Aug 2021 Preface This is a preliminary draft of Chapters 1–3 of our forthcoming textbook Introduction to cluster algebras, joint with Andrei Zelevinsky (1953–2013). Other chapters have been posted as arXiv:1707.07190 (Chapters 4–5), • arXiv:2008.09189 (Chapter 6), and • arXiv:2106.02160 (Chapter 7). • We expect to post additional chapters in the not so distant future. This book grew from the ten lectures given by Andrei at the NSF CBMS conference on Cluster Algebras and Applications at North Carolina State University in June 2006. The material of his lectures is much expanded but we still follow the original plan aimed at giving an accessible introduction to the subject for a general mathematical audience. Since its inception in [23], the theory of cluster algebras has been actively developed in many directions. We do not attempt to give a comprehensive treatment of the many connections and applications of this young theory. Our choice of topics reflects our personal taste; much of the book is based on the work done by Andrei and ourselves. Comments and suggestions are welcome. Sergey Fomin Lauren Williams Partially supported by the NSF grants DMS-1049513, DMS-1361789, and DMS-1664722. 2020 Mathematics Subject Classification. Primary 13F60. © 2016–2021 by Sergey Fomin, Lauren Williams, and Andrei Zelevinsky Contents Chapter 1. Total positivity 1 §1.1. Totally positive matrices 1 §1.2. The Grassmannian of 2-planes in m-space 4 §1.3.
    [Show full text]
  • Chapter 2 Solving Linear Equations
    Chapter 2 Solving Linear Equations Po-Ning Chen, Professor Department of Computer and Electrical Engineering National Chiao Tung University Hsin Chu, Taiwan 30010, R.O.C. 2.1 Vectors and linear equations 2-1 • What is this course Linear Algebra about? Algebra (代數) The part of mathematics in which letters and other general sym- bols are used to represent numbers and quantities in formulas and equations. Linear Algebra (線性代數) To combine these algebraic symbols (e.g., vectors) in a linear fashion. So, we will not combine these algebraic symbols in a nonlinear fashion in this course! x x1x2 =1 x 1 Example of nonlinear equations for = x : 2 x1/x2 =1 x x x x 1 1 + 2 =4 Example of linear equations for = x : 2 x1 − x2 =1 2.1 Vectors and linear equations 2-2 The linear equations can always be represented as matrix operation, i.e., Ax = b. Hence, the central problem of linear algorithm is to solve a system of linear equations. x − 2y =1 1 −2 x 1 Example of linear equations. ⇔ = 2 3x +2y =11 32 y 11 • A linear equation problem can also be viewed as a linear combination prob- lem for column vectors (as referred by the textbook as the column picture view). In contrast, the original linear equations (the red-color one above) is referred by the textbook as the row picture view. 1 −2 1 Column picture view: x + y = 3 2 11 1 −2 We want to find the scalar coefficients of column vectors and to form 3 2 1 another vector .
    [Show full text]
  • A Constructive Arbitrary-Degree Kronecker Product Decomposition of Tensors
    A CONSTRUCTIVE ARBITRARY-DEGREE KRONECKER PRODUCT DECOMPOSITION OF TENSORS KIM BATSELIER AND NGAI WONG∗ Abstract. We propose the tensor Kronecker product singular value decomposition (TKPSVD) that decomposes a real k-way tensor A into a linear combination of tensor Kronecker products PR (d) (1) with an arbitrary number of d factors A = j=1 σj Aj ⊗ · · · ⊗ Aj . We generalize the matrix (i) Kronecker product to tensors such that each factor Aj in the TKPSVD is a k-way tensor. The algorithm relies on reshaping and permuting the original tensor into a d-way tensor, after which a polyadic decomposition with orthogonal rank-1 terms is computed. We prove that for many (1) (d) different structured tensors, the Kronecker product factors Aj ;:::; Aj are guaranteed to inherit this structure. In addition, we introduce the new notion of general symmetric tensors, which includes many different structures such as symmetric, persymmetric, centrosymmetric, Toeplitz and Hankel tensors. Key words. Kronecker product, structured tensors, tensor decomposition, TTr1SVD, general- ized symmetric tensors, Toeplitz tensor, Hankel tensor AMS subject classifications. 15A69, 15B05, 15A18, 15A23, 15B57 1. Introduction. Consider the singular value decomposition (SVD) of the fol- lowing 16 × 9 matrix A~ 0 1:108 −0:267 −1:192 −0:267 −1:192 −1:281 −1:192 −1:281 1:102 1 B 0:417 −1:487 −0:004 −1:487 −0:004 −1:418 −0:004 −1:418 −0:228C B C B−0:127 1:100 −1:461 1:100 −1:461 0:729 −1:461 0:729 0:940 C B C B−0:748 −0:243 0:387 −0:243 0:387 −1:241 0:387 −1:241 −1:853C B C B 0:417
    [Show full text]
  • Introduction to Cluster Algebras Sergey Fomin
    Joint Introductory Workshop MSRI, Berkeley, August 2012 Introduction to cluster algebras Sergey Fomin (University of Michigan) Main references Cluster algebras I–IV: J. Amer. Math. Soc. 15 (2002), with A. Zelevinsky; Invent. Math. 154 (2003), with A. Zelevinsky; Duke Math. J. 126 (2005), with A. Berenstein & A. Zelevinsky; Compos. Math. 143 (2007), with A. Zelevinsky. Y -systems and generalized associahedra, Ann. of Math. 158 (2003), with A. Zelevinsky. Total positivity and cluster algebras, Proc. ICM, vol. 2, Hyderabad, 2010. 2 Cluster Algebras Portal hhttp://www.math.lsa.umich.edu/˜fomin/cluster.htmli Links to: • >400 papers on the arXiv; • a separate listing for lecture notes and surveys; • conferences, seminars, courses, thematic programs, etc. 3 Plan 1. Basic notions 2. Basic structural results 3. Periodicity and Grassmannians 4. Cluster algebras in full generality Tutorial session (G. Musiker) 4 FAQ • Who is your target audience? • Can I avoid the calculations? • Why don’t you just use a blackboard and chalk? • Where can I get the slides for your lectures? • Why do I keep seeing different definitions for the same terms? 5 PART 1: BASIC NOTIONS Motivations and applications Cluster algebras: a class of commutative rings equipped with a particular kind of combinatorial structure. Motivation: algebraic/combinatorial study of total positivity and dual canonical bases in semisimple algebraic groups (G. Lusztig). Some contexts where cluster-algebraic structures arise: • Lie theory and quantum groups; • quiver representations; • Poisson geometry and Teichm¨uller theory; • discrete integrable systems. 6 Total positivity A real matrix is totally positive (resp., totally nonnegative) if all its minors are positive (resp., nonnegative).
    [Show full text]
  • ON the PROPERTIES of the EXCHANGE GRAPH of a CLUSTER ALGEBRA Michael Gekhtman, Michael Shapiro, and Alek Vainshtein 1. Main Defi
    Math. Res. Lett. 15 (2008), no. 2, 321–330 c International Press 2008 ON THE PROPERTIES OF THE EXCHANGE GRAPH OF A CLUSTER ALGEBRA Michael Gekhtman, Michael Shapiro, and Alek Vainshtein Abstract. We prove a conjecture about the vertices and edges of the exchange graph of a cluster algebra A in two cases: when A is of geometric type and when A is arbitrary and its exchange matrix is nondegenerate. In the second case we also prove that the exchange graph does not depend on the coefficients of A. Both conjectures were formulated recently by Fomin and Zelevinsky. 1. Main definitions and results A cluster algebra is an axiomatically defined commutative ring equipped with a distinguished set of generators (cluster variables). These generators are subdivided into overlapping subsets (clusters) of the same cardinality that are connected via sequences of birational transformations of a particular kind, called cluster transfor- mations. Transfomations of this kind can be observed in many areas of mathematics (Pl¨ucker relations, Somos sequences and Hirota equations, to name just a few exam- ples). Cluster algebras were initially introduced in [FZ1] to study total positivity and (dual) canonical bases in semisimple algebraic groups. The rapid development of the cluster algebra theory revealed relations between cluster algebras and Grassmannians, quiver representations, generalized associahedra, Teichm¨uller theory, Poisson geome- try and many other branches of mathematics, see [Ze] and references therein. In the present paper we prove some conjectures on the general structure of cluster algebras formulated by Fomin and Zelevinsky in [FZ3]. To state our results, we recall the definition of a cluster algebra; for details see [FZ1, FZ4].
    [Show full text]
  • Spectral Clustering and Multidimensional Scaling: a Unified View
    Spectral clustering and multidimensional scaling: a unified view Fran¸coisBavaud Section d’Informatique et de M´ethodes Math´ematiques Facult´edes Lettres, Universit´ede Lausanne CH-1015 Lausanne, Switzerland (To appear in the proccedings of the IFCS 2006 Conference: “Data Science and Classification”, Ljubljana, Slovenia, July 25 - 29, 2006) Abstract. Spectral clustering is a procedure aimed at partitionning a weighted graph into minimally interacting components. The resulting eigen-structure is de- termined by a reversible Markov chain, or equivalently by a symmetric transition matrix F . On the other hand, multidimensional scaling procedures (and factorial correspondence analysis in particular) consist in the spectral decomposition of a kernel matrix K. This paper shows how F and K can be related to each other through a linear or even non-linear transformation leaving the eigen-vectors invari- ant. As illustrated by examples, this circumstance permits to define a transition matrix from a similarity matrix between n objects, to define Euclidean distances between the vertices of a weighted graph, and to elucidate the “flow-induced” nature of spatial auto-covariances. 1 Introduction and main results Scalar products between features define similarities between objects, and re- versible Markov chains define weighted graphs describing a stationary flow. It is natural to expect flows and similarities to be related: somehow, the exchange of flows between objects should enhance their similarity, and tran- sitions should preferentially occur between similar states. This paper formalizes the above intuition by demonstrating in a general framework that the symmetric matrices K and F possess an identical eigen- structure, where K (kernel, equation (2)) is a measure of similarity, and F (symmetrized transition.
    [Show full text]
  • Orbital-Dependent Exchange-Correlation Functionals in Density-Functional Theory Realized by the FLAPW Method
    Orbital-dependent exchange-correlation functionals in density-functional theory realized by the FLAPW method Von der Fakultät für Mathematik, Informatik und Naturwissenschaen der RWTH Aachen University zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaen genehmigte Dissertation vorgelegt von Dipl.-Phys. Markus Betzinger aus Fröndenberg Berichter: Prof. Dr. rer. nat. Stefan Blügel Prof. Dr. rer. nat. Andreas Görling Prof. Dr. rer. nat. Carsten Honerkamp Tag der mündlichen Prüfung: Ô¥.Ôò.òýÔÔ Diese Dissertation ist auf den Internetseiten der Hochschulbibliothek online verfügbar. is document was typeset with LATEX. Figures were created using Gnuplot, InkScape, and VESTA. Das Unverständlichste am Universum ist im Grunde, dass wir es verstehen können. Albert Einstein Abstract In this thesis, we extended the applicability of the full-potential linearized augmented-plane- wave (FLAPW) method, one of the most precise, versatile and generally applicable elec- tronic structure methods for solids working within the framework of density-functional the- ory (DFT), to orbital-dependent functionals for the exchange-correlation (xc) energy. In con- trast to the commonly applied local-density approximation (LDA) and generalized gradient approximation (GGA) for the xc energy, orbital-dependent functionals depend directly on the Kohn-Sham (KS) orbitals and only indirectly on the density. Two dierent schemes that deal with orbital-dependent functionals, the KS and the gen- eralized Kohn-Sham (gKS) formalism, have been realized. While the KS scheme requires a local multiplicative xc potential, the gKS scheme allows for a non-local potential in the one- particle Schrödinger equations. Hybrid functionals, combining some amount of the orbital-dependent exact exchange en- ergy with local or semi-local functionals of the density, are implemented within the gKS scheme.
    [Show full text]
  • Preconditioners for Symmetrized Toeplitz and Multilevel Toeplitz Matrices∗
    PRECONDITIONERS FOR SYMMETRIZED TOEPLITZ AND MULTILEVEL TOEPLITZ MATRICES∗ J. PESTANAy Abstract. When solving linear systems with nonsymmetric Toeplitz or multilevel Toeplitz matrices using Krylov subspace methods, the coefficient matrix may be symmetrized. The precondi- tioned MINRES method can then be applied to this symmetrized system, which allows rigorous upper bounds on the number of MINRES iterations to be obtained. However, effective preconditioners for symmetrized (multilevel) Toeplitz matrices are lacking. Here, we propose novel ideal preconditioners, and investigate the spectra of the preconditioned matrices. We show how these preconditioners can be approximated and demonstrate their effectiveness via numerical experiments. Key words. Toeplitz matrix, multilevel Toeplitz matrix, symmetrization, preconditioning, Krylov subspace method AMS subject classifications. 65F08, 65F10, 15B05, 35R11 1. Introduction. Linear systems (1.1) Anx = b; n×n n where An 2 R is a Toeplitz or multilevel Toeplitz matrix, and b 2 R arise in a range of applications. These include the discretization of partial differential and integral equations, time series analysis, and signal and image processing [7, 27]. Additionally, demand for fast numerical methods for fractional diffusion problems| which have recently received significant attention|has renewed interest in the solution of Toeplitz and Toeplitz-like systems [10, 26, 31, 32, 46]. Preconditioned iterative methods are often used to solve systems of the form (1.1). When An is Hermitian, CG [18] and MINRES [29] can be applied, and their descrip- tive convergence rate bounds guide the construction of effective preconditioners [7, 27]. On the other hand, convergence rates of preconditioned iterative methods for non- symmetric Toeplitz matrices are difficult to describe.
    [Show full text]
  • Cluster Algebras
    Cluster Algebras Philipp Lampe December 4, 2013 2 Contents 1 Informal introduction 5 1.1 Sequences of Laurent polynomials . .5 1.2 Exercises . .7 2 What are cluster algebras? 9 2.1 Quivers and adjacency matrices . .9 2.2 Quiver mutation . 14 2.3 Cluster algebras attached to quivers . 24 2.4 Skew-symmetrizable matrices, ice quivers and cluster algebras . 29 2.5 Exercises . 31 3 Examples of cluster algebras 33 3.1 Sequences and Diophantine equations attached to cluster algebras . 33 3.2 The Kronecker cluster algebra . 40 3.3 Cluster algebras of type A . 44 3.4 Exercises . 50 4 The Laurent phenomenon 53 4.1 The proof of the Laurent phenomenon . 53 4.2 Exercises . 57 5 Solutions to exercises 59 Bibliography 63 3 4 CONTENTS Chapter 1 Informal introduction 1.1 Sequences of Laurent polynomials In the lecture we wish to give an introduction to Sergey Fomin and Andrei Zelevinsky’s theory of cluster algebras. Fomin and Zelevinsky have introduced and studied cluster algebras in a series of four influential articles [FZ, FZ2, BFZ3, FZ4] (one of which is coauthored with Arkady Berenstein). Although their initial motivation comes from Lie theory, the definition of a cluster algebra is very elementary. We will give the precise definition in Chapter 2, but to give the reader a first idea let a and b be two undeterminates and let us consider the map b + 1 F : (a, b) 7! b, . a Surprisingly, the non-trivial algebraic identity F5 = id holds. This equation has a long and colour- ful history.
    [Show full text]
  • RIMS-1650 Integrable Systems: the R-Matrix Approach by Michael Semenov-Tian-Shansky December 2008 RESEARCH INSTITUTE for MATHEMA
    RIMS-1650 Integrable Systems: the r-matrix Approach By Michael Semenov-Tian-Shansky December 2008 RESEARCH INSTITUTE FOR MATHEMATICAL SCIENCES KYOTO UNIVERSITY, Kyoto, Japan Integrable Systems: the r-matrix Approach Michael Semenov-Tian-Shansky Universite´ de Bourgogne UFR Sciences & Techniques 9 av- enue Alain Savary BP 47970 - 21078 Dijon Cedex FRANCE Current address: Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502 JAPAN 1991 Mathematics Subject Classification. Primary 37K30 Secondary 17B37 17B68 37K10 Key words and phrases. Lie groups, Lie algebras, Poisson geometry, Poisson Lie groups, Symplectic geometry, Integrable systems, classical Yang-Baxter equation, Virasoro algebra, Differential Galois Theory Abstract. These lectures cover the theory of classical r-matrices sat- isfying the classical Yang–Baxter equations and their basic applications in the theory of Integrable Systems as well as in geometry of Poisson Lie groups. Special attention is given to the factorization problems associ- ated with classical r-matrices, the theory of dressing transformations, the geometric theory of difference zero curvature equations. We also discuss the relations between the Virasoro algebra and the Poisson Lie groups. Contents Introduction 4 Lecture 1. Preliminaries: Poisson Brackets, Poisson and Symplectic Manifolds, Symplectic Leaves, Reduction 7 1.1. Poisson Manifolds 7 1.2. Lie–Poisson Brackets 9 1.3. Poisson and Hamilton Reduction 12 1.4. Cotangent Bundle of a Lie Group. 18 Lecture 2. The r-Matrix Method and the Main Theorem 22 2.1. Introduction 22 2.2. Lie Dialgebras and Involutivity Theorem 23 2.3. Factorization Theorem 25 2.4. Factorization Theorem and Hamiltonian Reduction 26 2.5. Classical Yang-Baxter Identity and the General Theory of Classical r-Matrices 30 2.6.
    [Show full text]
  • Numerical Matrix Analysis
    Copyright ©2009 by the Society for Industrial and Applied Mathematics This electronic version is for personal use and may not be duplicated or distributed. Numerical Matrix Analysis Mathematics Applied and Industrial for Society Buy this book from SIAM at www.ec-securehost.com/SIAM/OT113.html Copyright ©2009 by the Society for Industrial and Applied Mathematics This electronic version is for personal use and may not be duplicated or distributed. Mathematics Applied and Industrial for Society Buy this book from SIAM at www.ec-securehost.com/SIAM/OT113.html Copyright ©2009 by the Society for Industrial and Applied Mathematics This electronic version is for personal use and may not be duplicated or distributed. Numerical Matrix Analysis Linear Systems and LeastMathematics Squares Applied Ilse C. F. Ipsenand North Carolina State University Raleigh, NorthIndustrial Carolina for Society Society for Industrial and Applied Mathematics Philadelphia Buy this book from SIAM at www.ec-securehost.com/SIAM/OT113.html Copyright ©2009 by the Society for Industrial and Applied Mathematics This electronic version is for personal use and may not be duplicated or distributed. Copyright © 2009 by the Society for Industrial and Applied Mathematics 10 9 8 7 6 5 4 3 2 1 All rights reserved. Printed in the United States of America. No part of this book may be reproduced, stored, or transmitted in any manner without the writtenMathematics permission of the publisher. For information, write to the Society for Industrial and Applied Mathematics, 3600 Market Street, 6th Floor, Philadelphia, PA, 19104-2688 USA. Trademarked names may be used in this book withoutApplied the inclusion of a trademark symbol.
    [Show full text]