Linear Algebra
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Modeling, Inference and Clustering for Equivalence Classes of 3-D Orientations Chuanlong Du Iowa State University
Iowa State University Capstones, Theses and Graduate Theses and Dissertations Dissertations 2014 Modeling, inference and clustering for equivalence classes of 3-D orientations Chuanlong Du Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/etd Part of the Statistics and Probability Commons Recommended Citation Du, Chuanlong, "Modeling, inference and clustering for equivalence classes of 3-D orientations" (2014). Graduate Theses and Dissertations. 13738. https://lib.dr.iastate.edu/etd/13738 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Graduate Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. Modeling, inference and clustering for equivalence classes of 3-D orientations by Chuanlong Du A dissertation submitted to the graduate faculty in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Major: Statistics Program of Study Committee: Stephen Vardeman, Co-major Professor Daniel Nordman, Co-major Professor Dan Nettleton Huaiqing Wu Guang Song Iowa State University Ames, Iowa 2014 Copyright c Chuanlong Du, 2014. All rights reserved. ii DEDICATION I would like to dedicate this dissertation to my mother Jinyan Xu. She always encourages me to follow my heart and to seek my dreams. Her words have always inspired me and encouraged me to face difficulties and challenges I came across in my life. I would also like to dedicate this dissertation to my wife Lisha Li without whose support and help I would not have been able to complete this work. -
MATH 2370, Practice Problems
MATH 2370, Practice Problems Kiumars Kaveh Problem: Prove that an n × n complex matrix A is diagonalizable if and only if there is a basis consisting of eigenvectors of A. Problem: Let A : V ! W be a one-to-one linear map between two finite dimensional vector spaces V and W . Show that the dual map A0 : W 0 ! V 0 is surjective. Problem: Determine if the curve 2 2 2 f(x; y) 2 R j x + y + xy = 10g is an ellipse or hyperbola or union of two lines. Problem: Show that if a nilpotent matrix is diagonalizable then it is the zero matrix. Problem: Let P be a permutation matrix. Show that P is diagonalizable. Show that if λ is an eigenvalue of P then for some integer m > 0 we have λm = 1 (i.e. λ is an m-th root of unity). Hint: Note that P m = I for some integer m > 0. Problem: Show that if λ is an eigenvector of an orthogonal matrix A then jλj = 1. n Problem: Take a vector v 2 R and let H be the hyperplane orthogonal n n to v. Let R : R ! R be the reflection with respect to a hyperplane H. Prove that R is a diagonalizable linear map. Problem: Prove that if λ1; λ2 are distinct eigenvalues of a complex matrix A then the intersection of the generalized eigenspaces Eλ1 and Eλ2 is zero (this is part of the Spectral Theorem). 1 Problem: Let H = (hij) be a 2 × 2 Hermitian matrix. Use the Min- imax Principle to show that if λ1 ≤ λ2 are the eigenvalues of H then λ1 ≤ h11 ≤ λ2. -
Sometimes Only Square Matrices Can Be Diagonalized 19
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 52, October 1975 SOMETIMESONLY SQUARE MATRICESCAN BE DIAGONALIZED LAWRENCE S. LEVY ABSTRACT. It is proved that every SQUARE matrix over a serial ring is equivalent to some diagonal matrix, even though there are rectangular matri- ces over these rings which cannot be diagonalized. Two matrices A and B over a ring R are equivalent if B - PAQ for in- vertible matrices P and Q over R- Following R. B. Warfield, we will call a module serial if its submodules are totally ordered by inclusion; and we will call a ring R serial it R is a direct sum of serial left modules, and also a direct sum of serial right mod- ules. When R is artinian, these become the generalized uniserial rings of Nakayama LE-GJ- Note that, in serial rings, finitely generated ideals do not have to be principal; for example, let R be any full ring of lower triangular matrices over a field. Moreover, it is well known (and easily proved) that if aR + bR is a nonprincipal right ideal in any ring, the 1x2 matrix [a b] is not equivalent to a diagonal matrix. Thus serial rings do not have the property that every rectangular matrix can be diagonalized. 1. Background. Serial rings are clearly semiperfect (P/radR is semisim- ple artinian and idempotents can be lifted modulo radP). (1.1) Every indecomposable projective module over a semiperfect ring R is S eR for some primitive idempotent e of R. (See [M].) A double-headed arrow -» will denote an "onto" homomorphism. -
Arxiv:1508.00183V2 [Math.RA] 6 Aug 2015 Atclrb Ed Yasemigroup a by field
A THEOREM OF KAPLANSKY REVISITED HEYDAR RADJAVI AND BAMDAD R. YAHAGHI Abstract. We present a new and simple proof of a theorem due to Kaplansky which unifies theorems of Kolchin and Levitzki on triangularizability of semigroups of matrices. We also give two different extensions of the theorem. As a consequence, we prove the counterpart of Kolchin’s Theorem for finite groups of unipotent matrices over division rings. We also show that the counterpart of Kolchin’s Theorem over division rings of characteristic zero implies that of Kaplansky’s Theorem over such division rings. 1. Introduction The purpose of this short note is three-fold. We give a simple proof of Kaplansky’s Theorem on the (simultaneous) triangularizability of semi- groups whose members all have singleton spectra. Our proof, although not independent of Kaplansky’s, avoids the deeper group theoretical aspects present in it and in other existing proofs we are aware of. Also, this proof can be adjusted to give an affirmative answer to Kolchin’s Problem for finite groups of unipotent matrices over division rings. In particular, it follows from this proof that the counterpart of Kolchin’s Theorem over division rings of characteristic zero implies that of Ka- plansky’s Theorem over such division rings. We also present extensions of Kaplansky’s Theorem in two different directions. M arXiv:1508.00183v2 [math.RA] 6 Aug 2015 Let us fix some notation. Let ∆ be a division ring and n(∆) the algebra of all n × n matrices over ∆. The division ring ∆ could in particular be a field. By a semigroup S ⊆ Mn(∆), we mean a set of matrices closed under multiplication. -
The Intersection of Some Classical Equivalence Classes of Matrices Mark Alan Mills Iowa State University
Iowa State University Capstones, Theses and Retrospective Theses and Dissertations Dissertations 1999 The intersection of some classical equivalence classes of matrices Mark Alan Mills Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/rtd Part of the Mathematics Commons Recommended Citation Mills, Mark Alan, "The intersection of some classical equivalence classes of matrices " (1999). Retrospective Theses and Dissertations. 12153. https://lib.dr.iastate.edu/rtd/12153 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Retrospective Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. UMI films the text directly fi'om the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type of computer printer. The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleedthrough, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. Oversize materials (e.g., maps, drawings, charts) are reproduced by sectioning the original, beginning at the upper left-hand comer and continuing firom left to right in equal sections with small overlaps. -
A Review of Matrix Scaling and Sinkhorn's Normal Form for Matrices
A review of matrix scaling and Sinkhorn’s normal form for matrices and positive maps Martin Idel∗ Zentrum Mathematik, M5, Technische Universität München, 85748 Garching Abstract Given a nonnegative matrix A, can you find diagonal matrices D1, D2 such that D1 AD2 is doubly stochastic? The answer to this question is known as Sinkhorn’s theorem. It has been proved with a wide variety of methods, each presenting a variety of possible generalisations. Recently, generalisations such as to positive maps between matrix algebras have become more and more interesting for applications. This text gives a review of over 70 years of matrix scaling. The focus lies on the mathematical landscape surrounding the problem and its solution as well as the generalisation to positive maps and contains hardly any nontrivial unpublished results. Contents 1. Introduction3 2. Notation and Preliminaries4 3. Different approaches to equivalence scaling5 3.1. Historical remarks . .6 3.2. The logarithmic barrier function . .8 3.3. Nonlinear Perron-Frobenius theory . 12 3.4. Entropy optimisation . 14 3.5. Convex programming and dual problems . 17 3.6. Topological (non-constructive) approaches . 20 arXiv:1609.06349v1 [math.RA] 20 Sep 2016 3.7. Other ideas . 21 3.7.1. Geometric proofs . 22 3.7.2. Other direct convergence proofs . 23 ∗[email protected] 1 4. Equivalence scaling 24 5. Other scalings 27 5.1. Matrix balancing . 27 5.2. DAD scaling . 29 5.3. Matrix Apportionment . 33 5.4. More general matrix scalings . 33 6. Generalised approaches 35 6.1. Direct multidimensional scaling . 35 6.2. Log-linear models and matrices as vectors . -
Math 623: Matrix Analysis Final Exam Preparation
Mike O'Sullivan Department of Mathematics San Diego State University Spring 2013 Math 623: Matrix Analysis Final Exam Preparation The final exam has two parts, which I intend to each take one hour. Part one is on the material covered since the last exam: Determinants; normal, Hermitian and positive definite matrices; positive matrices and Perron's theorem. The problems will all be very similar to the ones in my notes, or in the accompanying list. For part two, you will write an essay on what I see as the fundamental theme of the course, the four equivalence relations on matrices: matrix equivalence, similarity, unitary equivalence and unitary similarity (See p. 41 in Horn 2nd Ed. He calls matrix equivalence simply \equivalence."). Imagine you are writing for a fellow master's student. Your goal is to explain to them the key ideas. Matrix equivalence: (1) Define it. (2) Explain the relationship with abstract linear transformations and change of basis. (3) We found a simple set of representatives for the equivalence classes: identify them and sketch the proof. Similarity: (1) Define it. (2) Explain the relationship with abstract linear transformations and change of basis. (3) We found a set of representatives for the equivalence classes using Jordan matrices. State the Jordan theorem. (4) The proof of the Jordan theorem can be broken into two parts: (1) writing the ambient space as a direct sum of generalized eigenspaces, (2) classifying nilpotent matrices. Sketch the proof of each. Explain the role of invariant spaces. Unitary equivalence (1) Define it. (2) Explain the relationship with abstract inner product spaces and change of basis. -
Chapter 6 Jordan Canonical Form
Chapter 6 Jordan Canonical Form 175 6.1 Primary Decomposition Theorem For a general matrix, we shall now try to obtain the analogues of the ideas we have in the case of diagonalizable matrices. Diagonalizable Case General Case Minimal Polynomial: Minimal Polynomial: k k Y rj Y rj mA(λ) = (λ − λj) mA(λ) = (λ − λj) j=1 j=1 where rj = 1 for all j where 1 ≤ rj ≤ aj for all j Eigenspaces: Eigenspaces: Wj = Null Space of (A − λjI) Wj = Null Space of (A − λjI) dim(Wj) = gj and gj = aj for all j dim(Wj) = gj and gj ≤ aj for all j Generalized Eigenspaces: rj Vj = Null Space of (A − λjI) dim(Vj) = aj Decomposition: Decomposition: n n F = W1 ⊕ W2 ⊕ Wj ⊕ · · · ⊕ Wk W = W1⊕W2⊕Wj⊕· · ·⊕Wk is a subspace of F Primary Decomposition Theorem: n F = V1 ⊕ V2 ⊕ Vj ⊕ Vk n n x 2 F =) 9 unique x 2 F =) 9 unique xj 2 Wj, 1 ≤ j ≤ k such that vj 2 Vj, 1 ≤ j ≤ k such that x = x1 + x2 + ··· + xj + ··· + xk x = v1 + v2 + ··· + vj + ··· + vk 176 6.2 Analogues of Lagrange Polynomial Diagonalizable Case General Case Define Define k k Y Y ri fj(λ) = (λ − λi) Fj(λ) = (λ − λi) i=1 i=1 i6=j i6=j This is a set of coprime polynomials This is a set of coprime polynomials and and hence there exist polynomials hence there exist polynomials Gj(λ), 1 ≤ gj(λ), 1 ≤ j ≤ k such that j ≤ k such that k k X X fj(λ)gj(λ) = 1 Fj(λ)Gj(λ) = 1 j=1 j=1 The polynomials The polynomials `j(λ) = fj(λ)gj(λ); 1 ≤ j ≤ k Lj(λ) = Fj(λ)Gj(λ); 1 ≤ j ≤ k are the Lagrange Interpolation are the counterparts of the polynomials Lagrange Interpolation polynomials `1(λ) + `2(λ) + ··· + `k(λ) = 1 L1(λ) + L2(λ) + ··· + Lk(λ) = 1 λ1`1(λ)+λ2`2(λ)+···+λk`k(λ) = λ λ1L1(λ) + λ2L2(λ) + ··· + λkLk(λ) = d(λ) (a polynomial but it need not be = λ) For i 6= j the product `i(λ)`j(λ) has For i 6= j the product Li(λ)Lj(λ) has a a factor mA (λ) factor mA (λ) 6.3 Matrix Decomposition In the diagonalizable case, using the Lagrange polynomials we obtained a decomposition of the matrix A. -
An Effective Lie–Kolchin Theorem for Quasi-Unipotent Matrices
Linear Algebra and its Applications 581 (2019) 304–323 Contents lists available at ScienceDirect Linear Algebra and its Applications www.elsevier.com/locate/laa An effective Lie–Kolchin Theorem for quasi-unipotent matrices Thomas Koberda a, Feng Luo b,∗, Hongbin Sun b a Department of Mathematics, University of Virginia, Charlottesville, VA 22904-4137, USA b Department of Mathematics, Rutgers University, Hill Center – Busch Campus, 110 Frelinghuysen Road, Piscataway, NJ 08854, USA a r t i c l e i n f oa b s t r a c t Article history: We establish an effective version of the classical Lie–Kolchin Received 27 April 2019 Theorem. Namely, let A, B ∈ GLm(C)be quasi-unipotent Accepted 17 July 2019 matrices such that the Jordan Canonical Form of B consists Available online 23 July 2019 of a single block, and suppose that for all k 0the Submitted by V.V. Sergeichuk matrix ABk is also quasi-unipotent. Then A and B have a A, B < C MSC: common eigenvector. In particular, GLm( )is a primary 20H20, 20F38 solvable subgroup. We give applications of this result to the secondary 20F16, 15A15 representation theory of mapping class groups of orientable surfaces. Keywords: © 2019 Elsevier Inc. All rights reserved. Lie–Kolchin theorem Unipotent matrices Solvable groups Mapping class groups * Corresponding author. E-mail addresses: [email protected] (T. Koberda), fl[email protected] (F. Luo), [email protected] (H. Sun). URLs: http://faculty.virginia.edu/Koberda/ (T. Koberda), http://sites.math.rutgers.edu/~fluo/ (F. Luo), http://sites.math.rutgers.edu/~hs735/ (H. -
A Local Approach to Matrix Equivalence*
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector A Local Approach to Matrix Equivalence* Larry J. Gerstein Department of Mathematics University of California Santa Barbara, California 93106 Submitted by Hans Schneider ABSTRACT Matrix equivalence over principal ideal domains is considered, using the tech- nique of localization from commutative algebra. This device yields short new proofs for a variety of results. (Some of these results were known earlier via the theory of determinantal divisors.) A new algorithm is presented for calculation of the Smith normal form of a matrix, and examples are included. Finally, the natural analogue of the Witt-Grothendieck ring for quadratic forms is considered in the context of matrix equivalence. INTRODUCTION It is well known that every matrix A over a principal ideal domain is equivalent to a diagonal matrix S(A) whose main diagonal entries s,(A),s,(A), . divide each other in succession. [The s,(A) are the invariant factors of A; they are unique up to unit multiples.] How should one effect this diagonalization? If the ring is Euclidean, a sequence of elementary row and column operations will do the job; but in the general case one usually relies on the theory of determinantal divisors: Set d,,(A) = 1, and for 1 < k < r = rankA define d,(A) to be the greatest common divisor of all the k x k subdeterminants of A; then +(A) = dk(A)/dk_-l(A) for 1 < k < r, and sk(A) = 0 if k > r. The calculation of invariant factors via determinantal divisors in this way is clearly a very tedious process. -
Dimensions of Nilpotent Algebras
DIMENSIONS OF NILPOTENT ALGEBRAS AMAJOR QUALIFYING PROJECT REPORT SUBMITTED TO THE FACULTY OF WORCESTER POLYTECHNIC INSTITUTE IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF BACHELOR OF SCIENCE WRITTEN BY KWABENA A. ADWETEWA-BADU APPROVED BY: MARCH 14, 2021 THIS REPORT REPRESENTS THE WORK OF WPI UNDERGRADUATE STUDENTS SUBMITTED TO THE FACULTY AS EVIDENCE OF COMPLETION OF A DEGREE REQUIREMENT. 1 Abstract In this thesis, we study algebras of nilpotent matrices. In the first chapter, we give complete proofs of some well-known results on the structure of a linear operator. In particular, we prove that every nilpotent linear transformation can be represented by a strictly upper triangular matrix. In Chapter 2, we generalise this result to algebras of nilpotent matrices. We prove the famous Lie-Kolchin theorem, which states that an algebra is nilpotent if and only if it is conjugate to an algebra of strictly upper triangular matrices. In Chapter 3 we introduce a family of algebras constructed from graphs and directed graphs. We characterise the directed graphs which give nilpotent algebras. To be precise, a digraph generates an algebra of nilpotency class d if and only if it is acyclic with no paths of length ≥ d. Finally, in Chapter 4, we give Jacobson’s proof of a Theorem of Schur on the maximal dimension of a subalgebra of Mn(k) of nilpotency class 2. We relate this result to problems in external graph theory, and give lower bounds on the dimension of subalgebras of nilpotency class d in Mn(k) for every pair of integers d and n. -
Jordan Canonical Form of a Nilpotent Matrix Math 422 Schur's Triangularization Theorem Tells Us That Every Matrix a Is Unitari
Jordan Canonical Form of a Nilpotent Matrix Math 422 Schur’s Triangularization Theorem tells us that every matrix A is unitarily similar to an upper triangular matrix T . However, the only thing certain at this point is that the the diagonal entries of T are the eigenvalues of A. The off-diagonal entries of T seem unpredictable and out of control. Recall that the Core-Nilpotent Decomposition of a singular matrix A of index k produces a block diagonal matrix C 0 0 L ∙ ¸ similar to A in which C is non-singular, rank (C)=rank Ak , and L is nilpotent of index k.Isitpossible to simplify C and L via similarity transformations and obtain triangular matrices whose off-diagonal entries are predictable? The goal of this lecture is to do exactly this¡ ¢ for nilpotent matrices. k 1 k Let L be an n n nilpotent matrix of index k. Then L − =0and L =0. Let’s compute the eigenvalues × k k 16 k 1 k 1 k 2 of L. Suppose x =0satisfies Lx = λx; then 0=L x = L − (Lx)=L − (λx)=λL − x = λL − (Lx)= 2 k 2 6 k λ L − x = = λ x; thus λ =0is the only eigenvalue of L. ··· 1 Now if L is diagonalizable, there is an invertible matrix P and a diagonal matrix D such that P − LP = D. Since the diagonal entries of D are the eigenvalues of L, and λ =0is the only eigenvalue of L,wehave 1 D =0. Solving P − LP =0for L gives L =0. Thus a diagonalizable nilpotent matrix is the zero matrix, or equivalently, a non-zero nilpotent matrix L is not diagonalizable.