Positivity Properties of the Matrix (I + J)I+J
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Stirling Matrix Via Pascal Matrix
LinearAlgebraanditsApplications329(2001)49–59 www.elsevier.com/locate/laa StirlingmatrixviaPascalmatrix Gi-SangCheon a,∗,Jin-SooKim b aDepartmentofMathematics,DaejinUniversity,Pocheon487-711,SouthKorea bDepartmentofMathematics,SungkyunkwanUniversity,Suwon440-746,SouthKorea Received11May2000;accepted17November2000 SubmittedbyR.A.Brualdi Abstract ThePascal-typematricesobtainedfromtheStirlingnumbersofthefirstkinds(n,k)and ofthesecondkindS(n,k)arestudied,respectively.Itisshownthatthesematricescanbe factorizedbythePascalmatrices.AlsotheLDU-factorizationofaVandermondematrixof theformVn(x,x+1,...,x+n−1)foranyrealnumberxisobtained.Furthermore,some well-knowncombinatorialidentitiesareobtainedfromthematrixrepresentationoftheStirling numbers,andthesematricesaregeneralizedinoneortwovariables.©2001ElsevierScience Inc.Allrightsreserved. AMSclassification:05A19;05A10 Keywords:Pascalmatrix;Stirlingnumber;Stirlingmatrix 1.Introduction Forintegersnandkwithnk0,theStirlingnumbersofthefirstkinds(n,k) andofthesecondkindS(n,k)canbedefinedasthecoefficientsinthefollowing expansionofavariablex(see[3,pp.271–279]): n n−k k [x]n = (−1) s(n,k)x k=0 and ∗ Correspondingauthor. E-mailaddresses:[email protected](G.-S.Cheon),[email protected](J.-S.Kim). 0024-3795/01/$-seefrontmatter2001ElsevierScienceInc.Allrightsreserved. PII:S0024-3795(01)00234-8 50 G.-S. Cheon, J.-S. Kim / Linear Algebra and its Applications 329 (2001) 49–59 n n x = S(n,k)[x]k, (1.1) k=0 where x(x − 1) ···(x − n + 1) if n 1, [x] = (1.2) n 1ifn = 0. It is known that for an n, k 0, the s(n,k), S(n,k) and [n]k satisfy the following Pascal-type recurrence relations: s(n,k) = s(n − 1,k− 1) + (n − 1)s(n − 1,k), S(n,k) = S(n − 1,k− 1) + kS(n − 1,k), (1.3) [n]k =[n − 1]k + k[n − 1]k−1, where s(n,0) = s(0,k)= S(n,0) = S(0,k)=[0]k = 0ands(0, 0) = S(0, 0) = 1, and moreover the S(n,k) satisfies the following formula known as ‘vertical’ recur- rence relation: n− 1 n − 1 S(n,k) = S(l,k − 1). -
Pascal Matrices Alan Edelman and Gilbert Strang Department of Mathematics, Massachusetts Institute of Technology [email protected] and [email protected]
Pascal Matrices Alan Edelman and Gilbert Strang Department of Mathematics, Massachusetts Institute of Technology [email protected] and [email protected] Every polynomial of degree n has n roots; every continuous function on [0, 1] attains its maximum; every real symmetric matrix has a complete set of orthonormal eigenvectors. “General theorems” are a big part of the mathematics we know. We can hardly resist the urge to generalize further! Remove hypotheses, make the theorem tighter and more difficult, include more functions, move into Hilbert space,. It’s in our nature. The other extreme in mathematics might be called the “particular case”. One specific function or group or matrix becomes special. It obeys the general rules, like everyone else. At the same time it has some little twist that connects familiar objects in a neat way. This paper is about an extremely particular case. The familiar object is Pascal’s triangle. The little twist begins by putting that triangle of binomial coefficients into a matrix. Three different matrices—symmetric, lower triangular, and upper triangular—can hold Pascal’s triangle in a convenient way. Truncation produces n by n matrices Sn and Ln and Un—the pattern is visible for n = 4: 1 1 1 1 1 1 1 1 1 1 2 3 4 1 1 1 2 3 S4 = L4 = U4 = . 1 3 6 10 1 2 1 1 3 1 4 10 20 1 3 3 1 1 We mention first a very specific fact: The determinant of every Sn is 1. (If we emphasized det Ln = 1 and det Un = 1, you would write to the Editor. -
Arxiv:1904.01037V3 [Math.GR]
AN EFFECTIVE LIE–KOLCHIN THEOREM FOR QUASI-UNIPOTENT MATRICES THOMAS KOBERDA, FENG LUO, AND HONGBIN SUN Abstract. We establish an effective version of the classical Lie–Kolchin Theo- rem. Namely, let A, B P GLmpCq be quasi–unipotent matrices such that the Jordan Canonical Form of B consists of a single block, and suppose that for all k ě 0 the matrix ABk is also quasi–unipotent. Then A and B have a common eigenvector. In particular, xA, Byă GLmpCq is a solvable subgroup. We give applications of this result to the representation theory of mapping class groups of orientable surfaces. 1. Introduction Let V be a finite dimensional vector space over an algebraically closed field. In this paper, we study the structure of certain subgroups of GLpVq which contain “sufficiently many” elements of a relatively simple form. We are motivated by the representation theory of the mapping class group of a surface of hyperbolic type. If S be an orientable surface of genus 2 or more, the mapping class group ModpS q is the group of homotopy classes of orientation preserving homeomor- phisms of S . The group ModpS q is generated by certain mapping classes known as Dehn twists, which are defined for essential simple closed curves of S . Here, an essential simple closed curve is a free homotopy class of embedded copies of S 1 in S which is homotopically essential, in that the homotopy class represents a nontrivial conjugacy class in π1pS q which is not the homotopy class of a boundary component or a puncture of S . -
Seminar VII for the Course GROUP THEORY in PHYSICS Micael Flohr
Seminar VII for the course GROUP THEORY IN PHYSICS Mic~ael Flohr The classical Lie groups 25. January 2005 MATRIX LIE GROUPS Most Lie groups one ever encouters in physics are realized as matrix Lie groups and thus as subgroups of GL(n, R) or GL(n, C). This is the group of invertibel n × n matrices with coefficients in R or C, respectively. This is a Lie group, since it forms as an open subset of the vector space of n × n matrices a manifold. Matrix multiplication is certainly a differentiable map, as is taking the inverse via Cramer’s rule. The only condition defining the open 2 subset is that the determinat must not be zero, which implies that dimKGL(n, K) = n is the same as the one of the vector space Mn(K). However, GL(n, R) is not connected, because we cannot move continuously from a matrix with determinant less than zero to one with determinant larger than zero. It is worth mentioning that gl(n, K) is the vector space of all n × n matrices over the field K, equipped with the standard commutator as Lie bracket. We can describe most other Lie groups as subgroups of GL(n, K) for either K = R or K = C. There are two ways to do so. Firstly, one can give restricting equations to the coefficients of the matrices. Secondly, one can find subgroups of the automorphisms of V =∼ Kn, which conserve a given structure on Kn. In the following, we give some examples for this: SL(n, K). -
Inertia of the Matrix [(Pi + Pj) ]
isid/ms/2013/12 October 20, 2013 http://www.isid.ac.in/estatmath/eprints r Inertia of the matrix [(pi + pj) ] Rajendra Bhatia and Tanvi Jain Indian Statistical Institute, Delhi Centre 7, SJSS Marg, New Delhi{110 016, India r INERTIA OF THE MATRIX [(pi + pj) ] RAJENDRA BHATIA* AND TANVI JAIN** Abstract. Let p1; : : : ; pn be positive real numbers. It is well r known that for every r < 0 the matrix [(pi + pj) ] is positive def- inite. Our main theorem gives a count of the number of positive and negative eigenvalues of this matrix when r > 0: Connections with some other matrices that arise in Loewner's theory of oper- ator monotone functions and in the theory of spline interpolation are discussed. 1. Introduction Let p1; p2; : : : ; pn be distinct positive real numbers. The n×n matrix 1 C = [ ] is known as the Cauchy matrix. The special case pi = i pi+pj 1 gives the Hilbert matrix H = [ i+j ]: Both matrices have been studied by several authors in diverse contexts and are much used as test matrices in numerical analysis. The Cauchy matrix is known to be positive definite. It possessesh ai ◦r 1 stronger property: for each r > 0 the entrywise power C = r (pi+pj ) is positive definite. (See [4] for a proof.) The object of this paper is to study positivity properties of the related family of matrices r Pr = [(pi + pj) ]; r ≥ 0: (1) The inertia of a Hermitian matrix A is the triple In(A) = (π(A); ζ(A); ν(A)) ; in which π(A); ζ(A) and ν(A) stand for the number of positive, zero, and negative eigenvalues of A; respectively. -
The Pascal Matrix Function and Its Applications to Bernoulli Numbers and Bernoulli Polynomials and Euler Numbers and Euler Polynomials
The Pascal Matrix Function and Its Applications to Bernoulli Numbers and Bernoulli Polynomials and Euler Numbers and Euler Polynomials Tian-Xiao He ∗ Jeff H.-C. Liao y and Peter J.-S. Shiue z Dedicated to Professor L. C. Hsu on the occasion of his 95th birthday Abstract A Pascal matrix function is introduced by Call and Velleman in [3]. In this paper, we will use the function to give a unified approach in the study of Bernoulli numbers and Bernoulli poly- nomials. Many well-known and new properties of the Bernoulli numbers and polynomials can be established by using the Pascal matrix function. The approach is also applied to the study of Euler numbers and Euler polynomials. AMS Subject Classification: 05A15, 65B10, 33C45, 39A70, 41A80. Key Words and Phrases: Pascal matrix, Pascal matrix func- tion, Bernoulli number, Bernoulli polynomial, Euler number, Euler polynomial. ∗Department of Mathematics, Illinois Wesleyan University, Bloomington, Illinois 61702 yInstitute of Mathematics, Academia Sinica, Taipei, Taiwan zDepartment of Mathematical Sciences, University of Nevada, Las Vegas, Las Vegas, Nevada, 89154-4020. This work is partially supported by the Tzu (?) Tze foundation, Taipei, Taiwan, and the Institute of Mathematics, Academia Sinica, Taipei, Taiwan. The author would also thank to the Institute of Mathematics, Academia Sinica for the hospitality. 1 2 T. X. He, J. H.-C. Liao, and P. J.-S. Shiue 1 Introduction A large literature scatters widely in books and journals on Bernoulli numbers Bn, and Bernoulli polynomials Bn(x). They can be studied by means of the binomial expression connecting them, n X n B (x) = B xn−k; n ≥ 0: (1) n k k k=0 The study brings consistent attention of researchers working in combi- natorics, number theory, etc. -
Q-Pascal and Q-Bernoulli Matrices, an Umbral Approach Thomas Ernst
U.U.D.M. Report 2008:23 q-Pascal and q-Bernoulli matrices, an umbral approach Thomas Ernst Department of Mathematics Uppsala University q- PASCAL AND q-BERNOULLI MATRICES, AN UMBRAL APPROACH THOMAS ERNST Abstract A q-analogue Hn,q ∈ Mat(n)(C(q)) of the Polya-Vein ma- trix is used to define the q-Pascal matrix. The Nalli–Ward–AlSalam (NWA) q-shift operator acting on polynomials is a commutative semi- group. The q-Cauchy-Vandermonde matrix generalizing Aceto-Trigiante is defined by the NWA q-shift operator. A new formula for a q-Cauchy- Vandermonde determinant with matrix elements equal to q-Ward num- bers is found. The matrix form of the q-derivatives of the q-Bernoulli polynomials can be expressed in terms of the Hn,q. With the help of a new q-matrix multiplication certain special q-analogues of Aceto- Trigiante and Brawer-Pirovino are found. The q-Cauchy-Vandermonde matrix can be expressed in terms of the q-Bernoulli matrix. With the help of the Jackson-Hahn-Cigler (JHC) q-Bernoulli polynomials, the q-analogue of the Bernoulli complementary argument theorem is ob- tained. Analogous results for q-Euler polynomials are obtained. The q-Pascal matrix is factorized by the summation matrices and the so- called q-unit matrices. 1. Introduction In this paper we are going to find q-analogues of matrix formulas from two pairs of authors: L. Aceto, & D. Trigiante [1], [2] and R. Brawer & M.Pirovino [5]. The umbral method of the author [11] is used to find natural q-analogues of the Pascal- and the Cauchy-Vandermonde matrices. -
Fast Approximation Algorithms for Cauchy Matrices, Polynomials and Rational Functions
City University of New York (CUNY) CUNY Academic Works Computer Science Technical Reports CUNY Academic Works 2013 TR-2013011: Fast Approximation Algorithms for Cauchy Matrices, Polynomials and Rational Functions Victor Y. Pan How does access to this work benefit ou?y Let us know! More information about this work at: https://academicworks.cuny.edu/gc_cs_tr/386 Discover additional works at: https://academicworks.cuny.edu This work is made publicly available by the City University of New York (CUNY). Contact: [email protected] Fast Approximation Algorithms for Cauchy Matrices, Polynomials and Rational Functions ? Victor Y. Pan Department of Mathematics and Computer Science Lehman College and the Graduate Center of the City University of New York Bronx, NY 10468 USA [email protected], home page: http://comet.lehman.cuny.edu/vpan/ Abstract. The papers [MRT05], [CGS07], [XXG12], and [XXCBa] have combined the advanced FMM techniques with transformations of matrix structures (traced back to [P90]) in order to devise numerically stable algorithms that approximate the solutions of Toeplitz, Hankel, Toeplitz- like, and Hankel-like linear systems of equations in nearly linear arith- metic time, versus classical cubic time and quadratic time of the previous advanced algorithms. We show that the power of these approximation al- gorithms can be extended to yield similar results for computations with other matrices that have displacement structure, which includes Van- dermonde and Cauchy matrices, as well as to polynomial and rational evaluation and interpolation. The resulting decrease of the running time of the known approximation algorithms is again by order of magnitude, from quadratic to nearly linear. -
Relation Between Lah Matrix and K-Fibonacci Matrix
International Journal of Mathematics Trends and Technology Volume 66 Issue 10, 116-122, October 2020 ISSN: 2231 – 5373 /doi:10.14445/22315373/IJMTT-V66I10P513 © 2020 Seventh Sense Research Group® Relation between Lah matrix and k-Fibonacci Matrix Irda Melina Zet#1, Sri Gemawati#2, Kartini Kartini#3 #Department of Mathematics, Faculty of Mathematics and Natural Sciences, University of Riau Bina Widya Campus, Pekanbaru 28293, Indonesia Abstract. - The Lah matrix is represented by 퐿푛, is a matrix where each entry is Lah number. Lah number is count the number of ways a set of n elements can be partitioned into k nonempty linearly ordered subsets. k-Fibonacci matrix, 퐹푛(푘) is a matrix which all the entries are k-Fibonacci numbers. k-Fibonacci numbers are consist of the first term being 0, the second term being 1 and the next term depends on a natural number k. In this paper, a new matrix is defined namely 퐴푛 where it is not commutative to multiplicity of two matrices, so that another matrix 퐵푛 is defined such that 퐴푛 ≠ 퐵푛. The result is two forms of factorization from those matrices. In addition, the properties of the relation of Lah matrix and k- Fibonacci matrix is yielded as well. Keywords — Lah numbers, Lah matrix, k-Fibonacci numbers, k-Fibonacci matrix. I. INTRODUCTION Guo and Qi [5] stated that Lah numbers were introduced in 1955 and discovered by Ivo. Lah numbers are count the number of ways a set of n elements can be partitioned into k non-empty linearly ordered subsets. Lah numbers [9] is denoted by 퐿(푛, 푘) for every 푛, 푘 are elements of integers with initial value 퐿(0,0) = 1. -
An Accurate Computational Solution of Totally Positive Cauchy Linear Systems
AN ACCURATE COMPUTATIONAL SOLUTION OF TOTALLY POSITIVE CAUCHY LINEAR SYSTEMS Abdoulaye Cissé Andrew Anda Computer Science Department Computer Science Department St. Cloud State University St. Cloud State University St. Cloud, MN 56301 St. Cloud, MN 56301 [email protected] [email protected] Abstract ≈ ’ ∆ 1 The Cauchy linear systems C(x, y)V = b where C(x, y) = ∆ , x = (xi ) , y = ( yi ) , « xi − y j V = (vi ) , b = (bi ) can be solved very accurately regardless of condition number using Björck-Pereyra-type methods. We explain this phenomenon by the following facts. First, the Björck-Pereyra methods, written in matrix form, are essentially an accurate and efficient bidiagonal decomposition of the inverse of the matrix, as obtained by Neville Elimination with no pivoting. Second, each nontrivial entry of this bidiagonal decomposition is a product of two quotients of (initial) minors. We use this method to derive and implement a new method of accurately and efficiently solving totally positive Cauchy linear systems in time complexity O(n2). Keywords: Accuracy, Björck-Pereyra-type methods, Cauchy linear systems, floating point architectures, generalized Vandermonde linear systems, Neville elimination, totally positive, LDU decomposition. Note: In this paper we assume that the nodes xi and yj are pair wise distinct and C is totally positive 1 Background Definition 1.1 A Cauchy matrix is defined as: ≈ ’ ∆ 1 C(x, y) = ∆ , for i, j = 1,2,..., n (1.1) « xi − y j where the nodes xi and yj are assumed pair wise distinct. C(x,y) is totally positive (TP) if 2 2 0 < y1 < y2 < < yn−1 < yn < x1 < x2 < < xn−1 < xn . -
Factorizations for Q-Pascal Matrices of Two Variables
Spec. Matrices 2015; 3:207–213 Research Article Open Access Thomas Ernst Factorizations for q-Pascal matrices of two variables DOI 10.1515/spma-2015-0020 Received July 15, 2015; accepted September 14, 2015 Abstract: In this second article on q-Pascal matrices, we show how the previous factorizations by the sum- mation matrices and the so-called q-unit matrices extend in a natural way to produce q-analogues of Pascal matrices of two variables by Z. Zhang and M. Liu as follows 0 ! 1n−1 i Φ x y xi−j yi+j n,q( , ) = @ j A . q i,j=0 We also find two different matrix products for 0 ! 1n−1 i j Ψ x y i j + xi−j yi+j n,q( , ; , ) = @ j A . q i,j=0 Keywords: q-Pascal matrix; q-unit matrix; q-matrix multiplication 1 Introduction Once upon a time, Brawer and Pirovino [2, p.15 (1), p. 16(3)] found factorizations of the Pascal matrix and its inverse by the summation and difference matrices. In another article [7] we treated q-Pascal matrices and the corresponding factorizations. It turns out that an analoguous reasoning can be used to find q-analogues of the two variable factorizations by Zhang and Liu [13]. The purpose of this paper is thus to continue the q-analysis- matrix theme from our earlier papers [3]-[4] and [6]. To this aim, we define two new kinds of q-Pascal matrices, the lower triangular Φn,q matrix and the Ψn,q, both of two variables. To be able to write down addition and subtraction formulas for the most important q-special functions, i.e. -
Euler Matrices and Their Algebraic Properties Revisited
Appl. Math. Inf. Sci. 14, No. 4, 1-14 (2020) 1 Applied Mathematics & Information Sciences An International Journal http://dx.doi.org/10.18576/amis/AMIS submission quintana ramirez urieles FINAL VERSION Euler Matrices and their Algebraic Properties Revisited Yamilet Quintana1,∗, William Ram´ırez 2 and Alejandro Urieles3 1 Department of Pure and Applied Mathematics, Postal Code: 89000, Caracas 1080 A, Simon Bol´ıvar University, Venezuela 2 Department of Natural and Exact Sciences, University of the Coast - CUC, Barranquilla, Colombia 3 Faculty of Basic Sciences - Mathematics Program, University of Atl´antico, Km 7, Port Colombia, Barranquilla, Colombia Received: 2 Sep. 2019, Revised: 23 Feb. 2020, Accepted: 13 Apr. 2020 Published online: 1 Jul. 2020 Abstract: This paper addresses the generalized Euler polynomial matrix E (α)(x) and the Euler matrix E . Taking into account some properties of Euler polynomials and numbers, we deduce product formulae for E (α)(x) and define the inverse matrix of E . We establish some explicit expressions for the Euler polynomial matrix E (x), which involves the generalized Pascal, Fibonacci and Lucas matrices, respectively. From these formulae, we get some new interesting identities involving Fibonacci and Lucas numbers. Also, we provide some factorizations of the Euler polynomial matrix in terms of Stirling matrices, as well as a connection between the shifted Euler matrices and Vandermonde matrices. Keywords: Euler polynomials, Euler matrix, generalized Euler matrix, generalized Pascal matrix, Fibonacci matrix,