5.5 the Inverse of a Matrix
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Introduction to Linear Bialgebra
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by University of New Mexico University of New Mexico UNM Digital Repository Mathematics and Statistics Faculty and Staff Publications Academic Department Resources 2005 INTRODUCTION TO LINEAR BIALGEBRA Florentin Smarandache University of New Mexico, [email protected] W.B. Vasantha Kandasamy K. Ilanthenral Follow this and additional works at: https://digitalrepository.unm.edu/math_fsp Part of the Algebra Commons, Analysis Commons, Discrete Mathematics and Combinatorics Commons, and the Other Mathematics Commons Recommended Citation Smarandache, Florentin; W.B. Vasantha Kandasamy; and K. Ilanthenral. "INTRODUCTION TO LINEAR BIALGEBRA." (2005). https://digitalrepository.unm.edu/math_fsp/232 This Book is brought to you for free and open access by the Academic Department Resources at UNM Digital Repository. It has been accepted for inclusion in Mathematics and Statistics Faculty and Staff Publications by an authorized administrator of UNM Digital Repository. For more information, please contact [email protected], [email protected], [email protected]. INTRODUCTION TO LINEAR BIALGEBRA W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Technology, Madras Chennai – 600036, India e-mail: [email protected] web: http://mat.iitm.ac.in/~wbv Florentin Smarandache Department of Mathematics University of New Mexico Gallup, NM 87301, USA e-mail: [email protected] K. Ilanthenral Editor, Maths Tiger, Quarterly Journal Flat No.11, Mayura Park, 16, Kazhikundram Main Road, Tharamani, Chennai – 600 113, India e-mail: [email protected] HEXIS Phoenix, Arizona 2005 1 This book can be ordered in a paper bound reprint from: Books on Demand ProQuest Information & Learning (University of Microfilm International) 300 N. -
Linear Algebra: Example Sheet 2 of 4
Michaelmas Term 2014 SJW Linear Algebra: Example Sheet 2 of 4 1. (Another proof of the row rank column rank equality.) Let A be an m × n matrix of (column) rank r. Show that r is the least integer for which A factorises as A = BC with B 2 Matm;r(F) and C 2 Matr;n(F). Using the fact that (BC)T = CT BT , deduce that the (column) rank of AT equals r. 2. Write down the three types of elementary matrices and find their inverses. Show that an n × n matrix A is invertible if and only if it can be written as a product of elementary matrices. Use this method to find the inverse of 0 1 −1 0 1 @ 0 0 1 A : 0 3 −1 3. Let A and B be n × n matrices over a field F . Show that the 2n × 2n matrix IB IB C = can be transformed into D = −A 0 0 AB by elementary row operations (which you should specify). By considering the determinants of C and D, obtain another proof that det AB = det A det B. 4. (i) Let V be a non-trivial real vector space of finite dimension. Show that there are no endomorphisms α; β of V with αβ − βα = idV . (ii) Let V be the space of infinitely differentiable functions R ! R. Find endomorphisms α; β of V which do satisfy αβ − βα = idV . 5. Find the eigenvalues and give bases for the eigenspaces of the following complex matrices: 0 1 1 0 1 0 1 1 −1 1 0 1 1 −1 1 @ 0 3 −2 A ; @ 0 3 −2 A ; @ −1 3 −1 A : 0 1 0 0 1 0 −1 1 1 The second and third matrices commute; find a basis with respect to which they are both diagonal. -
Determinants Linear Algebra MATH 2010
Determinants Linear Algebra MATH 2010 • Determinants can be used for a lot of different applications. We will look at a few of these later. First of all, however, let's talk about how to compute a determinant. • Notation: The determinant of a square matrix A is denoted by jAj or det(A). • Determinant of a 2x2 matrix: Let A be a 2x2 matrix a b A = c d Then the determinant of A is given by jAj = ad − bc Recall, that this is the same formula used in calculating the inverse of A: 1 d −b 1 d −b A−1 = = ad − bc −c a jAj −c a if and only if ad − bc = jAj 6= 0. We will later see that if the determinant of any square matrix A 6= 0, then A is invertible or nonsingular. • Terminology: For larger matrices, we need to use cofactor expansion to find the determinant of A. First of all, let's define a few terms: { Minor: A minor, Mij, of the element aij is the determinant of the matrix obtained by deleting the ith row and jth column. ∗ Example: Let 2 −3 4 2 3 A = 4 6 3 1 5 4 −7 −8 Then to find M11, look at element a11 = −3. Delete the entire column and row that corre- sponds to a11 = −3, see the image below. Then M11 is the determinant of the remaining matrix, i.e., 3 1 M11 = = −8(3) − (−7)(1) = −17: −7 −8 ∗ Example: Similarly, to find M22 can be found looking at the element a22 and deleting the same row and column where this element is found, i.e., deleting the second row, second column: Then −3 2 M22 = = −8(−3) − 4(−2) = 16: 4 −8 { Cofactor: The cofactor, Cij is given by i+j Cij = (−1) Mij: Basically, the cofactor is either Mij or −Mij where the sign depends on the location of the element in the matrix. -
8 Rank of a Matrix
8 Rank of a matrix We already know how to figure out that the collection (v1;:::; vk) is linearly dependent or not if each n vj 2 R . Recall that we need to form the matrix [v1 j ::: j vk] with the given vectors as columns and see whether the row echelon form of this matrix has any free variables. If these vectors linearly independent (this is an abuse of language, the correct phrase would be \if the collection composed of these vectors is linearly independent"), then, due to the theorems we proved, their span is a subspace of Rn of dimension k, and this collection is a basis of this subspace. Now, what if they are linearly dependent? Still, their span will be a subspace of Rn, but what is the dimension and what is a basis? Naively, I can answer this question by looking at these vectors one by one. In particular, if v1 =6 0 then I form B = (v1) (I know that one nonzero vector is linearly independent). Next, I add v2 to this collection. If the collection (v1; v2) is linearly dependent (which I know how to check), I drop v2 and take v3. If independent then I form B = (v1; v2) and add now third vector v3. This procedure will lead to the vectors that form a basis of the span of my original collection, and their number will be the dimension. Can I do it in a different, more efficient way? The answer if \yes." As a side result we'll get one of the most important facts of the basic linear algebra. -
Linear Algebra Review James Chuang
Linear algebra review James Chuang December 15, 2016 Contents 2.1 vector-vector products ............................................... 1 2.2 matrix-vector products ............................................... 2 2.3 matrix-matrix products ............................................... 4 3.2 the transpose .................................................... 5 3.3 symmetric matrices ................................................. 5 3.4 the trace ....................................................... 6 3.5 norms ........................................................ 6 3.6 linear independence and rank ............................................ 7 3.7 the inverse ...................................................... 7 3.8 orthogonal matrices ................................................. 8 3.9 range and nullspace of a matrix ........................................... 8 3.10 the determinant ................................................... 9 3.11 quadratic forms and positive semidefinite matrices ................................ 10 3.12 eigenvalues and eigenvectors ........................................... 11 3.13 eigenvalues and eigenvectors of symmetric matrices ............................... 12 4.1 the gradient ..................................................... 13 4.2 the Hessian ..................................................... 14 4.3 gradients and hessians of linear and quadratic functions ............................. 15 4.5 gradients of the determinant ............................................ 16 4.6 eigenvalues -
A New Algorithm to Obtain the Adjugate Matrix Using CUBLAS on GPU González, H.E., Carmona, L., J.J
ISSN: 2277-3754 ISO 9001:2008 Certified International Journal of Engineering and Innovative Technology (IJEIT) Volume 7, Issue 4, October 2017 A New Algorithm to Obtain the Adjugate Matrix using CUBLAS on GPU González, H.E., Carmona, L., J.J. Information Technology Department, ININ-ITTLA Information Technology Department, ININ proved to be the best option for most practical applications. Abstract— In this paper a parallel code for obtain the Adjugate The new transformation proposed here can be obtained from Matrix with real coefficients are used. We postulate a new linear it. Next, we briefly review this topic. transformation in matrix product form and we apply this linear transformation to an augmented matrix (A|I) by means of both a minimum and a complete pivoting strategies, then we obtain the II. LU-MATRICIAL DECOMPOSITION WITH GT. Adjugate matrix. Furthermore, if we apply this new linear NOTATION AND DEFINITIONS transformation and the above pivot strategy to a augmented The problem of solving a linear system of equations matrix (A|b), we obtain a Cramer’s solution of the linear system of Ax b is central to the field of matrix computation. There are equations. That new algorithm present an O n3 computational several ways to perform the elimination process necessary for complexity when n . We use subroutines of CUBLAS 2nd A, b R its matrix triangulation. We will focus on the Doolittle-Gauss rd and 3 levels in double precision and we obtain correct numeric elimination method: the algorithm of choice when A is square, results. dense, and un-structured. nxn Index Terms—Adjoint matrix, Adjugate matrix, Cramer’s Let us assume that AR is nonsingular and that we rule, CUBLAS, GPU. -
Field of U-Invariants of Adjoint Representation of the Group GL (N, K)
Field of U-invariants of adjoint representation of the group GL(n,K) K.A.Vyatkina A.N.Panov ∗ It is known that the field of invariants of an arbitrary unitriangular group is rational (see [1]). In this paper for adjoint representation of the group GL(n,K) we present the algebraically system of generators of the field of U-invariants. Note that the algorithm for calculating generators of the field of invariants of an arbitrary rational representation of an unipotent group was presented in [2, Chapter 1]. The invariants was constructed using induction on the length of Jordan-H¨older series. In our case the length is equal to n2; that is why it is difficult to apply the algorithm. −1 Let us consider the adjoint representation AdgA = gAg of the group GL(n,K), where K is a field of zero characteristic, on the algebra of matri- ces M = Mat(n,K). The adjoint representation determines the representation −1 ρg on K[M] (resp. K(M)) by formula ρgf(A)= f(g Ag). Let U be the subgroup of upper triangular matrices in GL(n,K) with units on the diagonal. A polynomial (rational function) f on M is called an U- invariant, if ρuf = f for every u ∈ U. The set of U-invariant rational functions K(M)U is a subfield of K(M). Let {xi,j} be a system of standard coordinate functions on M. Construct a X X∗ ∗ X X∗ X∗ X matrix = (xij). Let = (xij) be its adjugate matrix, · = · = det X · E. Denote by Jk the left lower corner minor of order k of the matrix X. -
Formalizing the Matrix Inversion Based on the Adjugate Matrix in HOL4 Liming Li, Zhiping Shi, Yong Guan, Jie Zhang, Hongxing Wei
Formalizing the Matrix Inversion Based on the Adjugate Matrix in HOL4 Liming Li, Zhiping Shi, Yong Guan, Jie Zhang, Hongxing Wei To cite this version: Liming Li, Zhiping Shi, Yong Guan, Jie Zhang, Hongxing Wei. Formalizing the Matrix Inversion Based on the Adjugate Matrix in HOL4. 8th International Conference on Intelligent Information Processing (IIP), Oct 2014, Hangzhou, China. pp.178-186, 10.1007/978-3-662-44980-6_20. hal-01383331 HAL Id: hal-01383331 https://hal.inria.fr/hal-01383331 Submitted on 18 Oct 2016 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. Distributed under a Creative Commons Attribution| 4.0 International License Formalizing the Matrix Inversion Based on the Adjugate Matrix in HOL4 Liming LI1, Zhiping SHI1?, Yong GUAN1, Jie ZHANG2, and Hongxing WEI3 1 Beijing Key Laboratory of Electronic System Reliability Technology, Capital Normal University, Beijing 100048, China [email protected], [email protected] 2 College of Information Science & Technology, Beijing University of Chemical Technology, Beijing 100029, China 3 School of Mechanical Engineering and Automation, Beihang University, Beijing 100083, China Abstract. This paper presents the formalization of the matrix inversion based on the adjugate matrix in the HOL4 system. -
Linearized Polynomials Over Finite Fields Revisited
Linearized polynomials over finite fields revisited∗ Baofeng Wu,† Zhuojun Liu‡ Abstract We give new characterizations of the algebra Ln(Fqn ) formed by all linearized polynomials over the finite field Fqn after briefly sur- veying some known ones. One isomorphism we construct is between ∨ L F n F F n n( q ) and the composition algebra qn ⊗Fq q . The other iso- morphism we construct is between Ln(Fqn ) and the so-called Dickson matrix algebra Dn(Fqn ). We also further study the relations between a linearized polynomial and its associated Dickson matrix, generalizing a well-known criterion of Dickson on linearized permutation polyno- mials. Adjugate polynomial of a linearized polynomial is then intro- duced, and connections between them are discussed. Both of the new characterizations can bring us more simple approaches to establish a special form of representations of linearized polynomials proposed re- cently by several authors. Structure of the subalgebra Ln(Fqm ) which are formed by all linearized polynomials over a subfield Fqm of Fqn where m|n are also described. Keywords Linearized polynomial; Composition algebra; Dickson arXiv:1211.5475v2 [math.RA] 1 Jan 2013 matrix algebra; Representation. ∗Partially supported by National Basic Research Program of China (2011CB302400). †Key Laboratory of Mathematics Mechanization, AMSS, Chinese Academy of Sciences, Beijing 100190, China. Email: [email protected] ‡Key Laboratory of Mathematics Mechanization, AMSS, Chinese Academy of Sciences, Beijing 100190, China. Email: [email protected] 1 1 Introduction n Let Fq and Fqn be the finite fields with q and q elements respectively, where q is a prime or a prime power. -
Arxiv:2012.03523V3 [Math.NT] 16 May 2021 Favne Cetfi Optn Eerh(SR Spr Ft of Part (CM4)
WRONSKIAN´ ALGEBRA AND BROADHURST–ROBERTS QUADRATIC RELATIONS YAJUN ZHOU To the memory of Dr. W (1986–2020) ABSTRACT. Throughalgebraicmanipulationson Wronskian´ matrices whose entries are reducible to Bessel moments, we present a new analytic proof of the quadratic relations conjectured by Broadhurst and Roberts, along with some generalizations. In the Wronskian´ framework, we reinterpret the de Rham intersection pairing through polynomial coefficients in Vanhove’s differential operators, and compute the Betti inter- section pairing via linear sum rules for on-shell and off-shell Feynman diagrams at threshold momenta. From the ideal generated by Broadhurst–Roberts quadratic relations, we derive new non-linear sum rules for on-shell Feynman diagrams, including an infinite family of determinant identities that are compatible with Deligne’s conjectures for critical values of motivic L-functions. CONTENTS 1. Introduction 1 2. W algebra 5 2.1. Wronskian´ matrices and Vanhove operators 5 2.2. Block diagonalization of on-shell Wronskian´ matrices 8 2.3. Sumrulesforon-shellandoff-shellBesselmoments 11 3. W⋆ algebra 20 3.1. Cofactors of Wronskians´ 21 3.2. Threshold behavior of Wronskian´ cofactors 24 3.3. Quadratic relations for off-shell Bessel moments 30 4. Broadhurst–Roberts quadratic relations 37 4.1. Quadratic relations for on-shell Bessel moments 37 4.2. Arithmetic applications to Feynman diagrams 43 4.3. Alternative representations of de Rham matrices 49 References 53 arXiv:2012.03523v3 [math.NT] 16 May 2021 Date: May 18, 2021. Keywords: Bessel moments, Feynman integrals, Wronskian´ matrices, Bernoulli numbers Subject Classification (AMS 2020): 11B68, 33C10, 34M35 (Primary) 81T18, 81T40 (Secondary) * This research was supported in part by the Applied Mathematics Program within the Department of Energy (DOE) Office of Advanced Scientific Computing Research (ASCR) as part of the Collaboratory on Mathematics for Mesoscopic Modeling of Materials (CM4). -
Lecture 5: Matrix Operations: Inverse
Lecture 5: Matrix Operations: Inverse • Inverse of a matrix • Computation of inverse using co-factor matrix • Properties of the inverse of a matrix • Inverse of special matrices • Unit Matrix • Diagonal Matrix • Orthogonal Matrix • Lower/Upper Triangular Matrices 1 Matrix Inverse • Inverse of a matrix can only be defined for square matrices. • Inverse of a square matrix exists only if the determinant of that matrix is non-zero. • Inverse matrix of 퐴 is noted as 퐴−1. • 퐴퐴−1 = 퐴−1퐴 = 퐼 • Example: 2 −1 0 1 • 퐴 = , 퐴−1 = , 1 0 −1 2 2 −1 0 1 0 1 2 −1 1 0 • 퐴퐴−1 = = 퐴−1퐴 = = 1 0 −1 2 −1 2 1 0 0 1 2 Inverse of a 3 x 3 matrix (using cofactor matrix) • Calculating the inverse of a 3 × 3 matrix is: • Compute the matrix of minors for A. • Compute the cofactor matrix by alternating + and – signs. • Compute the adjugate matrix by taking a transpose of cofactor matrix. • Divide all elements in the adjugate matrix by determinant of matrix 퐴. 1 퐴−1 = 푎푑푗(퐴) det(퐴) 3 Inverse of a 3 x 3 matrix (using cofactor matrix) 3 0 2 퐴 = 2 0 −2 0 1 1 0 −2 2 −2 2 0 1 1 0 1 0 1 2 2 2 0 2 3 2 3 0 Matrix of Minors = = −2 3 3 1 1 0 1 0 1 0 2 3 2 3 0 0 −10 0 0 −2 2 −2 2 0 2 2 2 1 −1 1 2 −2 2 Cofactor of A (퐂) = −2 3 3 .∗ −1 1 −1 = 2 3 −3 0 −10 0 1 −1 1 0 10 0 2 2 0 adj A = CT = −2 3 10 2 −3 0 2 2 0 0.2 0.2 0 1 1 A-1 = ∗ adj A = −2 3 10 = −0.2 0.3 1 |퐴| 10 2 −3 0 0.2 −0.3 0 4 Properties of Inverse of a Matrix • (A-1)-1 = A • (AB)-1 = B-1A-1 • (kA)-1 = k-1A-1 where k is a non-zero scalar. -
Contents 2 Matrices and Systems of Linear Equations
Business Math II (part 2) : Matrices and Systems of Linear Equations (by Evan Dummit, 2019, v. 1.00) Contents 2 Matrices and Systems of Linear Equations 1 2.1 Systems of Linear Equations . 1 2.1.1 Elimination, Matrix Formulation . 2 2.1.2 Row-Echelon Form and Reduced Row-Echelon Form . 4 2.1.3 Gaussian Elimination . 6 2.2 Matrix Operations: Addition and Multiplication . 8 2.3 Determinants and Inverses . 11 2.3.1 The Inverse of a Matrix . 11 2.3.2 The Determinant of a Matrix . 13 2.3.3 Cofactor Expansions and the Adjugate . 16 2.3.4 Matrices and Systems of Linear Equations, Revisited . 18 2 Matrices and Systems of Linear Equations In this chapter, we will discuss the problem of solving systems of linear equations, reformulate the problem using matrices, and then give the general procedure for solving such systems. We will then study basic matrix operations (addition and multiplication) and discuss the inverse of a matrix and how it relates to another quantity known as the determinant. We will then revisit systems of linear equations after reformulating them in the language of matrices. 2.1 Systems of Linear Equations • Our motivating problem is to study the solutions to a system of linear equations, such as the system x + 3x = 5 1 2 : 3x1 + x2 = −1 ◦ Recall that a linear equation is an equation of the form a1x1 + a2x2 + ··· + anxn = b, for some constants a1; : : : an; b, and variables x1; : : : ; xn. ◦ When we seek to nd the solutions to a system of linear equations, this means nding all possible values of the variables such that all equations are satised simultaneously.