Manin Matrices and Talalaev's Formula
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PSEUDO SCHUR COMPLEMENTS, PSEUDO PRINCIPAL PIVOT TRANSFORMS and THEIR INHERITANCE PROPERTIES∗ 1. Introduction. Let M ∈ R M×
Electronic Journal of Linear Algebra ISSN 1081-3810 A publication of the International Linear Algebra Society Volume 30, pp. 455-477, August 2015 ELA PSEUDO SCHUR COMPLEMENTS, PSEUDO PRINCIPAL PIVOT TRANSFORMS AND THEIR INHERITANCE PROPERTIES∗ K. BISHT† , G. RAVINDRAN‡, AND K.C. SIVAKUMAR§ Abstract. Extensions of the Schur complement and the principal pivot transform, where the usual inverses are replaced by the Moore-Penrose inverse, are revisited. These are called the pseudo Schur complement and the pseudo principal pivot transform, respectively. First, a generalization of the characterization of a block matrix to be an M-matrix is extended to the nonnegativity of the Moore-Penrose inverse. A comprehensive treatment of the fundamental properties of the extended notion of the principal pivot transform is presented. Inheritance properties with respect to certain matrix classes are derived, thereby generalizing some of the existing results. Finally, a thorough discussion on the preservation of left eigenspaces by the pseudo principal pivot transformation is presented. Key words. Principal pivot transform, Schur complement, Nonnegative Moore-Penrose inverse, P†-Matrix, R†-Matrix, Left eigenspace, Inheritance properties. AMS subject classifications. 15A09, 15A18, 15B48. 1. Introduction. Let M ∈ Rm×n be a block matrix partitioned as A B , C D where A ∈ Rk×k is nonsingular. Then the classical Schur complement of A in M denoted by M/A is given by D − CA−1B ∈ R(m−k)×(n−k). This notion has proved to be a fundamental idea in many applications like numerical analysis, statistics and operator inequalities, to name a few. We refer the reader to [27] for a comprehen- sive account of the Schur complement. -
Notes on the Schur Complement
University of Pennsylvania ScholarlyCommons Departmental Papers (CIS) Department of Computer & Information Science 12-10-2010 Notes on the Schur Complement Jean H. Gallier University of Pennsylvania, [email protected] Follow this and additional works at: https://repository.upenn.edu/cis_papers Part of the Computer Sciences Commons Recommended Citation Jean H. Gallier, "Notes on the Schur Complement", . December 2010. Gallier, J., Notes on the Schur Complement This paper is posted at ScholarlyCommons. https://repository.upenn.edu/cis_papers/601 For more information, please contact [email protected]. Notes on the Schur Complement Disciplines Computer Sciences Comments Gallier, J., Notes on the Schur Complement This working paper is available at ScholarlyCommons: https://repository.upenn.edu/cis_papers/601 The Schur Complement and Symmetric Positive Semidefinite (and Definite) Matrices Jean Gallier December 10, 2010 1 Schur Complements In this note, we provide some details and proofs of some results from Appendix A.5 (especially Section A.5.5) of Convex Optimization by Boyd and Vandenberghe [1]. Let M be an n × n matrix written a as 2 × 2 block matrix AB M = ; CD where A is a p × p matrix and D is a q × q matrix, with n = p + q (so, B is a p × q matrix and C is a q × p matrix). We can try to solve the linear system AB x c = ; CD y d that is Ax + By = c Cx + Dy = d; by mimicking Gaussian elimination, that is, assuming that D is invertible, we first solve for y getting y = D−1(d − Cx) and after substituting this expression for y in the first equation, we get Ax + B(D−1(d − Cx)) = c; that is, (A − BD−1C)x = c − BD−1d: 1 If the matrix A − BD−1C is invertible, then we obtain the solution to our system x = (A − BD−1C)−1(c − BD−1d) y = D−1(d − C(A − BD−1C)−1(c − BD−1d)): The matrix, A − BD−1C, is called the Schur Complement of D in M. -
A Complete Bibliography of Publications in Linear Algebra and Its Applications: 2010–2019
A Complete Bibliography of Publications in Linear Algebra and its Applications: 2010{2019 Nelson H. F. Beebe University of Utah Department of Mathematics, 110 LCB 155 S 1400 E RM 233 Salt Lake City, UT 84112-0090 USA Tel: +1 801 581 5254 FAX: +1 801 581 4148 E-mail: [email protected], [email protected], [email protected] (Internet) WWW URL: http://www.math.utah.edu/~beebe/ 12 March 2021 Version 1.74 Title word cross-reference KY14, Rim12, Rud12, YHH12, vdH14]. 24 [KAAK11]. 2n − 3[BCS10,ˇ Hil13]. 2 × 2 [CGRVC13, CGSCZ10, CM14, DW11, DMS10, JK11, KJK13, MSvW12, Yan14]. (−1; 1) [AAFG12].´ (0; 1) 2 × 2 × 2 [Ber13b]. 3 [BBS12b, NP10, Ghe14a]. (2; 2; 0) [BZWL13, Bre14, CILL12, CKAC14, Fri12, [CI13, PH12]. (A; B) [PP13b]. (α, β) GOvdD14, GX12a, Kal13b, KK14, YHH12]. [HW11, HZM10]. (C; λ; µ)[dMR12].(`; m) p 3n2 − 2 2n3=2 − 3n [MR13]. [DFG10]. (H; m)[BOZ10].(κ, τ) p 3n2 − 2 2n3=23n [MR14a]. 3 × 3 [CSZ10, CR10c]. (λ, 2) [BBS12b]. (m; s; 0) [Dru14, GLZ14, Sev14]. 3 × 3 × 2 [Ber13b]. [GH13b]. (n − 3) [CGO10]. (n − 3; 2; 1) 3 × 3 × 3 [BH13b]. 4 [Ban13a, BDK11, [CCGR13]. (!) [CL12a]. (P; R)[KNS14]. BZ12b, CK13a, FP14, NSW13, Nor14]. 4 × 4 (R; S )[Tre12].−1 [LZG14]. 0 [AKZ13, σ [CJR11]. 5 Ano12-30, CGGS13, DLMZ14, Wu10a]. 1 [BH13b, CHY12, KRH14, Kol13, MW14a]. [Ano12-30, AHL+14, CGGS13, GM14, 5 × 5 [BAD09, DA10, Hil12a, Spe11]. 5 × n Kal13b, LM12, Wu10a]. 1=n [CNPP12]. [CJR11]. 6 1 <t<2 [Seo14]. 2 [AIS14, AM14, AKA13, [DK13c, DK11, DK12a, DK13b, Kar11a]. -
The Schur Complement and H-Matrix Theory
UNIVERSITY OF NOVI SAD FACULTY OF TECHNICAL SCIENCES Maja Nedović THE SCHUR COMPLEMENT AND H-MATRIX THEORY DOCTORAL DISSERTATION Novi Sad, 2016 УНИВЕРЗИТЕТ У НОВОМ САДУ ФАКУЛТЕТ ТЕХНИЧКИХ НАУКА 21000 НОВИ САД, Трг Доситеја Обрадовића 6 КЉУЧНА ДОКУМЕНТАЦИЈСКА ИНФОРМАЦИЈА Accession number, ANO: Identification number, INO: Document type, DT: Monographic publication Type of record, TR: Textual printed material Contents code, CC: PhD thesis Author, AU: Maja Nedović Mentor, MN: Professor Ljiljana Cvetković, PhD Title, TI: The Schur complement and H-matrix theory Language of text, LT: English Language of abstract, LA: Serbian, English Country of publication, CP: Republic of Serbia Locality of publication, LP: Province of Vojvodina Publication year, PY: 2016 Publisher, PB: Author’s reprint Publication place, PP: Novi Sad, Faculty of Technical Sciences, Trg Dositeja Obradovića 6 Physical description, PD: 6/161/85/1/0/12/0 (chapters/pages/ref./tables/pictures/graphs/appendixes) Scientific field, SF: Applied Mathematics Scientific discipline, SD: Numerical Mathematics Subject/Key words, S/KW: H-matrices, Schur complement, Eigenvalue localization, Maximum norm bounds for the inverse matrix, Closure of matrix classes UC Holding data, HD: Library of the Faculty of Technical Sciences, Trg Dositeja Obradovića 6, Novi Sad Note, N: Abstract, AB: This thesis studies subclasses of the class of H-matrices and their applications, with emphasis on the investigation of the Schur complement properties. The contributions of the thesis are new nonsingularity results, bounds for the maximum norm of the inverse matrix, closure properties of some matrix classes under taking Schur complements, as well as results on localization and separation of the eigenvalues of the Schur complement based on the entries of the original matrix. -
Manin Matrices, Casimir Elements and Sugawara Operators
Manin matrices, Casimir elements and Sugawara operators Alexander Molev University of Sydney 1 I Origins and motivations. I Basic properties of Manin matrices. I Applications: I Casimir elements for gln. I Segal–Sugawara vectors for gln. Plan 2 I Basic properties of Manin matrices. I Applications: I Casimir elements for gln. I Segal–Sugawara vectors for gln. Plan I Origins and motivations. 2 I Applications: I Casimir elements for gln. I Segal–Sugawara vectors for gln. Plan I Origins and motivations. I Basic properties of Manin matrices. 2 I Segal–Sugawara vectors for gln. Plan I Origins and motivations. I Basic properties of Manin matrices. I Applications: I Casimir elements for gln. 2 Plan I Origins and motivations. I Basic properties of Manin matrices. I Applications: I Casimir elements for gln. I Segal–Sugawara vectors for gln. 2 By the seminal works of V. Drinfeld (1985) and M. Jimbo (1985), the universal enveloping algebra U(g) of a simple Lie algebra g admits a deformation Uq(g) in the class of Hopf algebras. The dual Hopf algebras are quantized algebras of functions Funq(G) on the associated Lie group G [N. Reshetikhin, L. Takhtajan and L. Faddeev 1990]. A detailed review of the theory and applications: V. Chari and A. Pressley, A guide to quantum groups, 1994. Quantum groups 3 The dual Hopf algebras are quantized algebras of functions Funq(G) on the associated Lie group G [N. Reshetikhin, L. Takhtajan and L. Faddeev 1990]. A detailed review of the theory and applications: V. Chari and A. Pressley, A guide to quantum groups, 1994. -
Scrambled Linear Pseudorandom Number Generators∗
Scrambled Linear Pseudorandom Number Generators∗ DAVID BLACKMAN, Independent researcher, Australia SEBASTIANO VIGNA, Università degli Studi di Milano, Italy Linear pseudorandom number generators are very popular due to their high speed, to the ease with which generators with a sizable state space can be created, and to their provable theoretical properties. However, they suffer from linear artifacts which show as failures in linearity-related statistical tests such as the binary-rank and the linear-complexity test. In this paper, we give three new contributions. First, we introduce two new linear transformations that have been handcrafted to have good statistical properties and at the same time to be programmable very efficiently on superscalar processors, or even directly in hardware. Then, we describe a new test for Hamming-weight dependencies that is able to discover subtle, previously unknown biases in existing generators (in particular, in linear ones). Finally, we describe a number of scramblers, that is, nonlinear functions applied to the state array that reduce or delete the linear artifacts, and propose combinations of linear transformations and scramblers that give extremely fast pseudorandom generators of high quality. A novelty in our approach is that we use ideas from the theory of filtered linear-feedback shift register to prove some properties of our scramblers, rather than relying purely on heuristics. In the end, we provide simple, extremely fast generators that use a few hundred bits of memory, have provable properties and pass very strong statistical tests. CCS Concepts: • Mathematics of computing → Random number generation; Additional Key Words and Phrases: Pseudorandom number generators 1 INTRODUCTION In the last twenty years, in particular since the introduction of the Mersennne Twister [34], linear1 pseudorandom generators have been very popular: indeed, they are often the stock generator provided by several programming languages. -
The Multivariate Normal Distribution
Multivariate normal distribution Linear combinations and quadratic forms Marginal and conditional distributions The multivariate normal distribution Patrick Breheny September 2 Patrick Breheny University of Iowa Likelihood Theory (BIOS 7110) 1 / 31 Multivariate normal distribution Linear algebra background Linear combinations and quadratic forms Definition Marginal and conditional distributions Density and MGF Introduction • Today we will introduce the multivariate normal distribution and attempt to discuss its properties in a fairly thorough manner • The multivariate normal distribution is by far the most important multivariate distribution in statistics • It’s important for all the reasons that the one-dimensional Gaussian distribution is important, but even more so in higher dimensions because many distributions that are useful in one dimension do not easily extend to the multivariate case Patrick Breheny University of Iowa Likelihood Theory (BIOS 7110) 2 / 31 Multivariate normal distribution Linear algebra background Linear combinations and quadratic forms Definition Marginal and conditional distributions Density and MGF Inverse • Before we get to the multivariate normal distribution, let’s review some important results from linear algebra that we will use throughout the course, starting with inverses • Definition: The inverse of an n × n matrix A, denoted A−1, −1 −1 is the matrix satisfying AA = A A = In, where In is the n × n identity matrix. • Note: We’re sort of getting ahead of ourselves by saying that −1 −1 A is “the” matrix satisfying -
1. Introduction
Pr´e-Publica¸c~oesdo Departamento de Matem´atica Universidade de Coimbra Preprint Number 17{07 ON SKEW-SYMMETRIC MATRICES RELATED TO THE VECTOR CROSS PRODUCT IN R7 P. D. BEITES, A. P. NICOLAS´ AND JOSE´ VITORIA´ Abstract: A study of real skew-symmetric matrices of orders 7 and 8, defined through the vector cross product in R7, is presented. More concretely, results on matrix properties, eigenvalues, (generalized) inverses and rotation matrices are established. Keywords: vector cross product, skew-symmetric matrix, matrix properties, eigen- values, (generalized) inverses, rotation matrices. Math. Subject Classification (2010): 15A72, 15B57, 15A18, 15A09, 15B10. 1. Introduction A classical result known as the generalized Hurwitz Theorem asserts that, over a field of characteristic different from 2, if A is a finite dimensional composition algebra with identity, then its dimension is equal to 1; 2; 4 or 8. Furthermore, A is isomorphic either to the base field, a separable quadratic extension of the base field, a generalized quaternion algebra or a generalized octonion algebra, [8]. A consequence of the cited theorem is that the values of n for which the Euclidean spaces Rn can be equippped with a binary vector cross product, satisfying the same requirements as the usual one in R3, are restricted to 1 (trivial case), 3 and 7. A complete account on the existence of r-fold vector cross products for d-dimensional vector spaces, where they are used to construct exceptional Lie superalgebras, is in [3]. The interest in octonions, seemingly forgotten for some time, resurged in the last decades, not only for their intrinsic mathematical relevance but also because of their applications, as well as those of the vector cross product in R7. -
Manin Matrices, Quantum Elliptic Commutative Families and Characteristic Polynomial of Elliptic Gaudin Model Vladimir Roubtsov, Alexey Silantyev, Dmitri Talalaev
Manin Matrices, Quantum Elliptic Commutative Families and Characteristic Polynomial of Elliptic Gaudin Model Vladimir Roubtsov, Alexey Silantyev, Dmitri Talalaev To cite this version: Vladimir Roubtsov, Alexey Silantyev, Dmitri Talalaev. Manin Matrices, Quantum Elliptic Commu- tative Families and Characteristic Polynomial of Elliptic Gaudin Model. Symmetry, Integrability and Geometry : Methods and Applications, National Academy of Science of Ukraine, 2009, 5 (110), Non spécifié. 10.3842/SIGMA.2009.110. hal-03054058 HAL Id: hal-03054058 https://hal.univ-angers.fr/hal-03054058 Submitted on 11 Dec 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Symmetry, Integrability and Geometry: Methods and Applications SIGMA 5 (2009), 110, 22 pages Manin Matrices, Quantum Elliptic Commutative Families and Characteristic Polynomial of Elliptic Gaudin Model? Vladimir RUBTSOV †§, Alexey SILANTYEV ‡ and Dmitri TALALAEV § † LAREMA, Universit´ed’Angers, 2 Boulevard Lavoisier, 49045 Angers, France E-mail: [email protected] URL: http://www.math.univ-angers.fr/∼volodya/ ‡ Department of Mathematics, University Gardens, University of Glasgow, G12 8QW, UK E-mail: [email protected] § ITEP, B. Cheremushkinskaja 25, 117218 Moscow, Russia E-mail: [email protected] Received March 30, 2009, in final form December 12, 2009; Published online December 24, 2009 doi:10.3842/SIGMA.2009.110 Abstract. -
Limits of Gaudin Algebras, Quantization of Bending Flows
ITEP-TH-78/06 LIMITS OF GAUDIN ALGEBRAS, QUANTIZATION OF BENDING FLOWS, JUCYS–MURPHY ELEMENTS AND GELFAND–TSETLIN BASES A. CHERVOV, G. FALQUI, AND L. RYBNIKOV Abstract. Gaudin algebras form a family of maximal commutative subalgebras in the tensor product of n copies of the universal enveloping algebra U(g) of a semisimple Lie algebra g. This family is parameterized by collections ⊗n of pairwise distinct complex numbers z1,...,zn . We obtain some new commutative subalgebras in U(g) as limit cases of Gaudin subalgebras. These commutative subalgebras turn to be related to the hamiltonians of bending flows and to the Gelfand–Tsetlin bases. We use this to prove the simplicity of spectrum in the Gaudin model for some new cases. Key words. Gaudin model, Bethe ansatz, bending flows, Gelfand–Tsetlin bases. Contents 1. Introduction 1 2. Preliminaries 3 2.1. Construction of Gaudin subalgebras. 3 2.2. Quantum Mishchenko-Fomenko ”shift of argument” subalgebras. 4 2.3. Talalaev’s formula. 4 2.4. The generators of A(z1,...,zn). 4 3. Limits of Gaudin algebras 6 4. Quantum hamiltonians of bending flows 7 5. Schur–Weyl duality and Jucys–Murphy elements 8 6. (glN , glM ) duality and Gelfand–Tsetlin algebra 8 7. Simplicity of the joint spectrum of the Gaudin algebras in the glN case 9 References 10 arXiv:0710.4971v2 [math.QA] 24 Nov 2007 1. Introduction Gaudin model was introduced in [G76] as a spin model related to the Lie algebra sl2 , and generalized to the case of arbitrary semisimple Lie algebras in [G83], 13.2.2. -
[Math.QA] 24 Jan 2003 Ihagbacsprrushwvr Hr R Nsm Sense Some in Are There However, Supergroups Algebraic with En Anyte Pwt Hsclproblems
ON ALGEBRAIC SUPERGROUPS AND QUANTUM DEFORMATIONS R. FIORESI* Dipartimento di Matematica, Universita’ di Bologna Piazza Porta San Donato 5, 40126 Bologna, Italy e-mail: fi[email protected] Abstract. We give the definitions of affine algebraic supervariety and affine algebraic supergroup through the functor of points and we relate them to the other definitions present in the literature. We study in detail the algebraic supergroups GL(m|n) and SL(m|n) and give explicitly the Hopf algebra structure of the algebra representing the functors of points. At the end we give also the quantization of GL(m|n) together with its coaction on suitable quantum spaces according to Manin’s philosophy. 1. Introduction The mathematical foundations of supergeometry were laid in the 60s by Berezin in [Be] and later by Leites [Le] Kostant [Ko] and Manin [Ma1] among many others, its origins being mainly tied up with physical problems. A new attention to the subject came later with the study of quantum fields and superstring. In the 1999 “Notes on Supersymmetry” Deligne and Morgan [DM] give a categorical point of view on supersymmetry notions developed originally by physicists and known from a more “operational” point of view. In the current definitions of supermanifold, the points of a supermanifold are points of an usual manifold and the adjective super refers to an additional structure on the structural sheaf of functions on the manifold. This sheaf is assumed to be a sheaf of commutative superalgebras, where a superalgebra is a Z2-graded algebra. When dealing with algebraic supergroups however, there are in some sense true points. -
Noncommutative Determinants, Cauchy–Binet Formulae, and Capelli-Type Identities
Noncommutative determinants, Cauchy–Binet formulae, and Capelli-type identities I. Generalizations of the Capelli and Turnbull identities Sergio Caracciolo Dipartimento di Fisica and INFN Universit`adegli Studi di Milano via Celoria 16 I-20133 Milano, ITALY [email protected] Alan D. Sokal∗ Department of Physics New York University 4 Washington Place New York, NY 10003 USA [email protected] Andrea Sportiello Dipartimento di Fisica and INFN Universit`adegli Studi di Milano via Celoria 16 I-20133 Milano, ITALY [email protected] Submitted: Sep 20, 2008; Accepted: Aug 3, 2009; Published: Aug 7, 2009 Mathematics Subject Classification: 15A15 (Primary); 05A19, 05A30, 05E15, 13A50, 15A24, 15A33, 15A72, 17B35, 20G05 (Secondary). Abstract We prove, by simple manipulation of commutators, two noncommutative gener- alizations of the Cauchy–Binet formula for the determinant of a product. As special cases we obtain elementary proofs of the Capelli identity from classical invariant theory and of Turnbull’s Capelli-type identities for symmetric and antisymmetric matrices. Key Words: Determinant, noncommutative determinant, row-determinant, column- determinant, Cauchy–Binet theorem, permanent, noncommutative ring, Capelli identity, Turnbull identity, Cayley identity, classical invariant theory, representation theory, Weyl algebra, right-quantum matrix, Cartier–Foata matrix, Manin matrix. ∗Also at Department of Mathematics, University College London, London WC1E 6BT, England. the electronic journal of combinatorics 16 (2009), #R103 1 1 Introduction n Let R be a commutative ring, and let A =(aij)i,j=1 be an n × n matrix with elements in R. Define as usual the determinant n det A := sgn(σ) aiσ(i) . (1.1) σ i=1 X∈Sn Y One of the first things one learns about the determinant is the multiplicative property: det(AB) = (det A)(det B) .