Applied Linear Algebra and Differential Equations
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Math 511 Advanced Linear Algebra Spring 2006
MATH 511 ADVANCED LINEAR ALGEBRA SPRING 2006 Sherod Eubanks HOMEWORK 2 x2:1 : 2; 5; 9; 12 x2:3 : 3; 6 x2:4 : 2; 4; 5; 9; 11 Section 2:1: Unitary Matrices Problem 2 If ¸ 2 σ(U) and U 2 Mn is unitary, show that j¸j = 1. Solution. If ¸ 2 σ(U), U 2 Mn is unitary, and Ux = ¸x for x 6= 0, then by Theorem 2:1:4(g), we have kxkCn = kUxkCn = k¸xkCn = j¸jkxkCn , hence j¸j = 1, as desired. Problem 5 Show that the permutation matrices in Mn are orthogonal and that the permutation matrices form a sub- group of the group of real orthogonal matrices. How many different permutation matrices are there in Mn? Solution. By definition, a matrix P 2 Mn is called a permutation matrix if exactly one entry in each row n and column is equal to 1, and all other entries are 0. That is, letting ei 2 C denote the standard basis n th element of C that has a 1 in the i row and zeros elsewhere, and Sn be the set of all permutations on n th elements, then P = [eσ(1) j ¢ ¢ ¢ j eσ(n)] = Pσ for some permutation σ 2 Sn such that σ(k) denotes the k member of σ. Observe that for any σ 2 Sn, and as ½ 1 if i = j eT e = σ(i) σ(j) 0 otherwise for 1 · i · j · n by the definition of ei, we have that 2 3 T T eσ(1)eσ(1) ¢ ¢ ¢ eσ(1)eσ(n) T 6 . -
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. -
Chapter Four Determinants
Chapter Four Determinants In the first chapter of this book we considered linear systems and we picked out the special case of systems with the same number of equations as unknowns, those of the form T~x = ~b where T is a square matrix. We noted a distinction between two classes of T ’s. While such systems may have a unique solution or no solutions or infinitely many solutions, if a particular T is associated with a unique solution in any system, such as the homogeneous system ~b = ~0, then T is associated with a unique solution for every ~b. We call such a matrix of coefficients ‘nonsingular’. The other kind of T , where every linear system for which it is the matrix of coefficients has either no solution or infinitely many solutions, we call ‘singular’. Through the second and third chapters the value of this distinction has been a theme. For instance, we now know that nonsingularity of an n£n matrix T is equivalent to each of these: ² a system T~x = ~b has a solution, and that solution is unique; ² Gauss-Jordan reduction of T yields an identity matrix; ² the rows of T form a linearly independent set; ² the columns of T form a basis for Rn; ² any map that T represents is an isomorphism; ² an inverse matrix T ¡1 exists. So when we look at a particular square matrix, the question of whether it is nonsingular is one of the first things that we ask. This chapter develops a formula to determine this. (Since we will restrict the discussion to square matrices, in this chapter we will usually simply say ‘matrix’ in place of ‘square matrix’.) More precisely, we will develop infinitely many formulas, one for 1£1 ma- trices, one for 2£2 matrices, etc. -
EUCLIDEAN DISTANCE MATRIX COMPLETION PROBLEMS June 6
EUCLIDEAN DISTANCE MATRIX COMPLETION PROBLEMS HAW-REN FANG∗ AND DIANNE P. O’LEARY† June 6, 2010 Abstract. A Euclidean distance matrix is one in which the (i, j) entry specifies the squared distance between particle i and particle j. Given a partially-specified symmetric matrix A with zero diagonal, the Euclidean distance matrix completion problem (EDMCP) is to determine the unspecified entries to make A a Euclidean distance matrix. We survey three different approaches to solving the EDMCP. We advocate expressing the EDMCP as a nonconvex optimization problem using the particle positions as variables and solving using a modified Newton or quasi-Newton method. To avoid local minima, we develop a randomized initial- ization technique that involves a nonlinear version of the classical multidimensional scaling, and a dimensionality relaxation scheme with optional weighting. Our experiments show that the method easily solves the artificial problems introduced by Mor´e and Wu. It also solves the 12 much more difficult protein fragment problems introduced by Hen- drickson, and the 6 larger protein problems introduced by Grooms, Lewis, and Trosset. Key words. distance geometry, Euclidean distance matrices, global optimization, dimensional- ity relaxation, modified Cholesky factorizations, molecular conformation AMS subject classifications. 49M15, 65K05, 90C26, 92E10 1. Introduction. Given the distances between each pair of n particles in Rr, n r, it is easy to determine the relative positions of the particles. In many applications,≥ though, we are given only some of the distances and we would like to determine the missing distances and thus the particle positions. We focus in this paper on algorithms to solve this distance completion problem. -
Rotation Matrix - Wikipedia, the Free Encyclopedia Page 1 of 22
Rotation matrix - Wikipedia, the free encyclopedia Page 1 of 22 Rotation matrix From Wikipedia, the free encyclopedia In linear algebra, a rotation matrix is a matrix that is used to perform a rotation in Euclidean space. For example the matrix rotates points in the xy -Cartesian plane counterclockwise through an angle θ about the origin of the Cartesian coordinate system. To perform the rotation, the position of each point must be represented by a column vector v, containing the coordinates of the point. A rotated vector is obtained by using the matrix multiplication Rv (see below for details). In two and three dimensions, rotation matrices are among the simplest algebraic descriptions of rotations, and are used extensively for computations in geometry, physics, and computer graphics. Though most applications involve rotations in two or three dimensions, rotation matrices can be defined for n-dimensional space. Rotation matrices are always square, with real entries. Algebraically, a rotation matrix in n-dimensions is a n × n special orthogonal matrix, i.e. an orthogonal matrix whose determinant is 1: . The set of all rotation matrices forms a group, known as the rotation group or the special orthogonal group. It is a subset of the orthogonal group, which includes reflections and consists of all orthogonal matrices with determinant 1 or -1, and of the special linear group, which includes all volume-preserving transformations and consists of matrices with determinant 1. Contents 1 Rotations in two dimensions 1.1 Non-standard orientation -
Dimensionality Reduction
CHAPTER 7 Dimensionality Reduction We saw in Chapter 6 that high-dimensional data has some peculiar characteristics, some of which are counterintuitive. For example, in high dimensions the center of the space is devoid of points, with most of the points being scattered along the surface of the space or in the corners. There is also an apparent proliferation of orthogonal axes. As a consequence high-dimensional data can cause problems for data mining and analysis, although in some cases high-dimensionality can help, for example, for nonlinear classification. Nevertheless, it is important to check whether the dimensionality can be reduced while preserving the essential properties of the full data matrix. This can aid data visualization as well as data mining. In this chapter we study methods that allow us to obtain optimal lower-dimensional projections of the data. 7.1 BACKGROUND Let the data D consist of n points over d attributes, that is, it is an n × d matrix, given as ⎛ ⎞ X X ··· Xd ⎜ 1 2 ⎟ ⎜ x x ··· x ⎟ ⎜x1 11 12 1d ⎟ ⎜ x x ··· x ⎟ D =⎜x2 21 22 2d ⎟ ⎜ . ⎟ ⎝ . .. ⎠ xn xn1 xn2 ··· xnd T Each point xi = (xi1,xi2,...,xid) is a vector in the ambient d-dimensional vector space spanned by the d standard basis vectors e1,e2,...,ed ,whereei corresponds to the ith attribute Xi . Recall that the standard basis is an orthonormal basis for the data T space, that is, the basis vectors are pairwise orthogonal, ei ej = 0, and have unit length ei = 1. T As such, given any other set of d orthonormal vectors u1,u2,...,ud ,withui uj = 0 T and ui = 1(orui ui = 1), we can re-express each point x as the linear combination x = a1u1 + a2u2 +···+adud (7.1) 183 184 Dimensionality Reduction T where the vector a = (a1,a2,...,ad ) represents the coordinates of x in the new basis. -
Alternating Sign Matrices, Extensions and Related Cones
See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/311671190 Alternating sign matrices, extensions and related cones Article in Advances in Applied Mathematics · May 2017 DOI: 10.1016/j.aam.2016.12.001 CITATIONS READS 0 29 2 authors: Richard A. Brualdi Geir Dahl University of Wisconsin–Madison University of Oslo 252 PUBLICATIONS 3,815 CITATIONS 102 PUBLICATIONS 1,032 CITATIONS SEE PROFILE SEE PROFILE Some of the authors of this publication are also working on these related projects: Combinatorial matrix theory; alternating sign matrices View project All content following this page was uploaded by Geir Dahl on 16 December 2016. The user has requested enhancement of the downloaded file. All in-text references underlined in blue are added to the original document and are linked to publications on ResearchGate, letting you access and read them immediately. Alternating sign matrices, extensions and related cones Richard A. Brualdi∗ Geir Dahly December 1, 2016 Abstract An alternating sign matrix, or ASM, is a (0; ±1)-matrix where the nonzero entries in each row and column alternate in sign, and where each row and column sum is 1. We study the convex cone generated by ASMs of order n, called the ASM cone, as well as several related cones and polytopes. Some decomposition results are shown, and we find a minimal Hilbert basis of the ASM cone. The notion of (±1)-doubly stochastic matrices and a generalization of ASMs are introduced and various properties are shown. For instance, we give a new short proof of the linear characterization of the ASM polytope, in fact for a more general polytope. -
Linear Algebra
Linear Algebra July 28, 2006 1 Introduction These notes are intended for use in the warm-up camp for incoming Berkeley Statistics graduate students. Welcome to Cal! We assume that you have taken a linear algebra course before and that most of the material in these notes will be a review of what you already know. If you have never taken such a course before, you are strongly encouraged to do so by taking math 110 (or the honors version of it), or by covering material presented and/or mentioned here on your own. If some of the material is unfamiliar, do not be intimidated! We hope you find these notes helpful! If not, you can consult the references listed at the end, or any other textbooks of your choice for more information or another style of presentation (most of the proofs on linear algebra part have been adopted from Strang, the proof of F-test from Montgomery et al, and the proof of bivariate normal density from Bickel and Doksum). Go Bears! 1 2 Vector Spaces A set V is a vector space over R and its elements are called vectors if there are 2 opera- tions defined on it: 1. Vector addition, that assigns to each pair of vectors v ; v V another vector w V 1 2 2 2 (we write v1 + v2 = w) 2. Scalar multiplication, that assigns to each vector v V and each scalar r R another 2 2 vector w V (we write rv = w) 2 that satisfy the following 8 conditions v ; v ; v V and r ; r R: 8 1 2 3 2 8 1 2 2 1. -
Linear Algebra with Exercises B
Linear Algebra with Exercises B Fall 2017 Kyoto University Ivan Ip These notes summarize the definitions, theorems and some examples discussed in class. Please refer to the class notes and reference books for proofs and more in-depth discussions. Contents 1 Abstract Vector Spaces 1 1.1 Vector Spaces . .1 1.2 Subspaces . .3 1.3 Linearly Independent Sets . .4 1.4 Bases . .5 1.5 Dimensions . .7 1.6 Intersections, Sums and Direct Sums . .9 2 Linear Transformations and Matrices 11 2.1 Linear Transformations . 11 2.2 Injection, Surjection and Isomorphism . 13 2.3 Rank . 14 2.4 Change of Basis . 15 3 Euclidean Space 17 3.1 Inner Product . 17 3.2 Orthogonal Basis . 20 3.3 Orthogonal Projection . 21 i 3.4 Orthogonal Matrix . 24 3.5 Gram-Schmidt Process . 25 3.6 Least Square Approximation . 28 4 Eigenvectors and Eigenvalues 31 4.1 Eigenvectors . 31 4.2 Determinants . 33 4.3 Characteristic polynomial . 36 4.4 Similarity . 38 5 Diagonalization 41 5.1 Diagonalization . 41 5.2 Symmetric Matrices . 44 5.3 Minimal Polynomials . 46 5.4 Jordan Canonical Form . 48 5.5 Positive definite matrix (Optional) . 52 5.6 Singular Value Decomposition (Optional) . 54 A Complex Matrix 59 ii Introduction Real life problems are hard. Linear Algebra is easy (in the mathematical sense). We make linear approximations to real life problems, and reduce the problems to systems of linear equations where we can then use the techniques from Linear Algebra to solve for approximate solutions. Linear Algebra also gives new insights and tools to the original problems. -
More Gaussian Elimination and Matrix Inversion
Week 7 More Gaussian Elimination and Matrix Inversion 7.1 Opening Remarks 7.1.1 Introduction * View at edX 289 Week 7. More Gaussian Elimination and Matrix Inversion 290 7.1.2 Outline 7.1. Opening Remarks..................................... 289 7.1.1. Introduction..................................... 289 7.1.2. Outline....................................... 290 7.1.3. What You Will Learn................................ 291 7.2. When Gaussian Elimination Breaks Down....................... 292 7.2.1. When Gaussian Elimination Works........................ 292 7.2.2. The Problem.................................... 297 7.2.3. Permutations.................................... 299 7.2.4. Gaussian Elimination with Row Swapping (LU Factorization with Partial Pivoting) 304 7.2.5. When Gaussian Elimination Fails Altogether................... 311 7.3. The Inverse Matrix.................................... 312 7.3.1. Inverse Functions in 1D.............................. 312 7.3.2. Back to Linear Transformations.......................... 312 7.3.3. Simple Examples.................................. 314 7.3.4. More Advanced (but Still Simple) Examples................... 320 7.3.5. Properties...................................... 324 7.4. Enrichment......................................... 325 7.4.1. Library Routines for LU with Partial Pivoting................... 325 7.5. Wrap Up.......................................... 327 7.5.1. Homework..................................... 327 7.5.2. Summary...................................... 327 7.1. Opening -
Pattern Avoidance in Alternating Sign Matrices
PATTERN AVOIDANCE IN ALTERNATING SIGN MATRICES ROBERT JOHANSSON AND SVANTE LINUSSON Abstract. We generalize the definition of a pattern from permu- tations to alternating sign matrices. The number of alternating sign matrices avoiding 132 is proved to be counted by the large Schr¨oder numbers, 1, 2, 6, 22, 90, 394 . .. We give a bijection be- tween 132-avoiding alternating sign matrices and Schr¨oder-paths, which gives a refined enumeration. We also show that the 132, 123- avoiding alternating sign matrices are counted by every second Fi- bonacci number. 1. Introduction For some time, we have emphasized the use of permutation matrices when studying pattern avoidance. This and the fact that alternating sign matrices are generalizations of permutation matrices, led to the idea of studying pattern avoidance in alternating sign matrices. The main results, concerning 132-avoidance, are presented in Section 3. In Section 5, we enumerate sets of alternating sign matrices avoiding two patterns of length three at once. In Section 6 we state some open problems. Let us start with the basic definitions. Given a permutation π of length n we will represent it with the n×n permutation matrix with 1 in positions (i, π(i)), 1 ≤ i ≤ n and 0 in all other positions. We will also suppress the 0s. 1 1 3 1 2 1 Figure 1. The permutation matrix of 132. In this paper, we will think of permutations in terms of permutation matrices. Definition 1.1. (Patterns in permutations) A permutation π is said to contain a permutation τ as a pattern if the permutation matrix of τ is a submatrix of π. -
Matrices That Commute with a Permutation Matrix
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Matrices That Commute With a Permutation Matrix Jeffrey L. Stuart Department of Mathematics University of Southern Mississippi Hattiesburg, Mississippi 39406-5045 and James R. Weaver Department of Mathematics and Statistics University of West Florida Pensacola. Florida 32514 Submitted by Donald W. Robinson ABSTRACT Let P be an n X n permutation matrix, and let p be the corresponding permuta- tion. Let A be a matrix such that AP = PA. It is well known that when p is an n-cycle, A is permutation similar to a circulant matrix. We present results for the band patterns in A and for the eigenstructure of A when p consists of several disjoint cycles. These results depend on the greatest common divisors of pairs of cycle lengths. 1. INTRODUCTION A central theme in matrix theory concerns what can be said about a matrix if it commutes with a given matrix of some special type. In this paper, we investigate the combinatorial structure and the eigenstructure of a matrix that commutes with a permutation matrix. In doing so, we follow a long tradition of work on classes of matrices that commute with a permutation matrix of a specified type. In particular, our work is related both to work on the circulant matrices, the results for which can be found in Davis’s compre- hensive book [5], and to work on the centrosymmetric matrices, which can be found in the book by Aitken [l] and in the papers by Andrew [2], Bamett, LZNEAR ALGEBRA AND ITS APPLICATIONS 150:255-265 (1991) 255 Q Elsevier Science Publishing Co., Inc., 1991 655 Avenue of the Americas, New York, NY 10010 0024-3795/91/$3.50 256 JEFFREY L.