DOCSLIB.ORG

Lecture 5: Matrix Operations: Inverse

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
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. • (AT)-1 = (A-1)T 5 Inverse of Identity matrices • Inverse of identity matrix is itself. • Because: • 퐼퐼 = 퐼 • Example: 1 2 1 0 1 2 • 퐴퐼 = 퐴, = 3 4 0 1 3 4 1 0 1 2 1 2 • 퐼−1퐴 = 퐴, = 0 1 3 4 3 4 Inverse of Diagonal matrices • The determinant of a diagonal matrix is the product of its diagonal elements. • If they all are non-zero, then determinant is non-zero and the matrix is invertible. • The inverse of a diagonal matrix A is another diagonal matrix B whose diagonal elements are the reciprocals of the diagonal elements of A. • Example: 2 0 0 • 퐴 = 0 1 0 , 퐴 = 2 × 1 × −1 = −2 (≠ 0) 0 0 −1 1 0 0 −1 2 • 퐴 = 0 1 0 0 0 −1 7 Inverse of Orthonormal matrices • Earlier, we saw that multiplication of an orthogonal (orthonormal) matrix in its transpose results in identity matrix • If 퐴 is an orthonormal matrix, its inverse is equal to its transpose • 퐴 is an orthonormal 푛 × 푛 matrix 푇 • Recall 퐴퐴 = 퐼푛 , where 퐼푛 is a 푛 × 푛 identity matrix • So, 퐴푇 = 퐴−1 8 Inverse of Upper/Lower Triangular Matrices • Inverse of an upper/lower triangular matrix is another upper/lower triangular matrix. • Inverse exists only if none of the diagonal element is zero. • Can be computed from first principles: Using the definition of an Inverse. 퐴퐴−1 = 퐼. No need to compute determinant. • Diagonal elements of 퐴−1is the reciprocal of the elements of 퐴. • Other elements are iteratively computed such that the product of the matrices is 퐼. Inverse of Upper/Lower Triangular Matrices Upper Triangular Matrix: 2 1 0.5 푋 2 1 0.5 푋 1 0 퐴 = ; 퐴−1 ; 0 −1 0 −1 0 −1 0 −1 0 1 푆표푙푣푖푛푔 푓표푟 푋 푤푒 푔푒푡 푋 = 0.5 Lower Triangular Matrix: 2 0 0.5 0 2 0 0.5 0 1 0 퐵 = ; 퐵−1 = ; = 2 1 푋 1 2 1 푋 1 0 1 푆표푙푣푖푛푔 푓표푟 푋 푤푒 푔푒푡 푋 = −1.
Load more

Recommended publications
  • 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.
    [Show full text]
  • The Inverse Along a Lower Triangular Matrix∗
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Universidade do Minho: RepositoriUM The inverse along a lower triangular matrix∗ Xavier Marya, Pedro Patr´ıciob aUniversit´eParis-Ouest Nanterre { La D´efense,Laboratoire Modal'X, 200 avenuue de la r´epublique,92000 Nanterre, France. email: [email protected] bDepartamento de Matem´aticae Aplica¸c~oes,Universidade do Minho, 4710-057 Braga, Portugal. email: [email protected] Abstract In this paper, we investigate the recently defined notion of inverse along an element in the context of matrices over a ring. Precisely, we study the inverse of a matrix along a lower triangular matrix, under some conditions. Keywords: Generalized inverse, inverse along an element, Dedekind-finite ring, Green's relations, rings AMS classification: 15A09, 16E50 1 Introduction In this paper, R is a ring with identity. We say a is (von Neumann) regular in R if a 2 aRa.A particular solution to axa = a is denoted by a−, and the set of all such solutions is denoted by af1g. Given a−; a= 2 af1g then x = a=aa− satisfies axa = a; xax = a simultaneously. Such a solution is called a reflexive inverse, and is denoted by a+. The set of all reflexive inverses of a is denoted by af1; 2g. Finally, a is group invertible if there is a# 2 af1; 2g that commutes with a, and a is Drazin invertible if ak is group invertible, for some non-negative integer k. This is equivalent to the existence of aD 2 R such that ak+1aD = ak; aDaaD = aD; aaD = aDa.
    [Show full text]
  • LU Decompositions We Seek a Factorization of a Square Matrix a Into the Product of Two Matrices Which Yields an Efficient Method
    LU Decompositions We seek a factorization of a square matrix A into the product of two matrices which yields an efficient method for solving the system where A is the coefficient matrix, x is our variable vector and is a constant vector for . The factorization of A into the product of two matrices is closely related to Gaussian elimination. Definition 1. A square matrix is said to be lower triangular if for all . 2. A square matrix is said to be unit lower triangular if it is lower triangular and each . 3. A square matrix is said to be upper triangular if for all . Examples 1. The following are all lower triangular matrices: , , 2. The following are all unit lower triangular matrices: , , 3. The following are all upper triangular matrices: , , We note that the identity matrix is only square matrix that is both unit lower triangular and upper triangular. Example Let . For elementary matrices (See solv_lin_equ2.pdf) , , and we find that . Now, if , then direct computation yields and . It follows that and, hence, that where L is unit lower triangular and U is upper triangular. That is, . Observe the key fact that the unit lower triangular matrix L “contains” the essential data of the three elementary matrices , , and . Definition We say that the matrix A has an LU decomposition if where L is unit lower triangular and U is upper triangular. We also call the LU decomposition an LU factorization. Example 1. and so has an LU decomposition. 2. The matrix has more than one LU decomposition. Two such LU factorizations are and .
    [Show full text]
  • Handout 9 More Matrix Properties; the Transpose
    Handout 9 More matrix properties; the transpose Square matrix properties These properties only apply to a square matrix, i.e. n £ n. ² The leading diagonal is the diagonal line consisting of the entries a11, a22, a33, . ann. ² A diagonal matrix has zeros everywhere except the leading diagonal. ² The identity matrix I has zeros o® the leading diagonal, and 1 for each entry on the diagonal. It is a special case of a diagonal matrix, and A I = I A = A for any n £ n matrix A. ² An upper triangular matrix has all its non-zero entries on or above the leading diagonal. ² A lower triangular matrix has all its non-zero entries on or below the leading diagonal. ² A symmetric matrix has the same entries below and above the diagonal: aij = aji for any values of i and j between 1 and n. ² An antisymmetric or skew-symmetric matrix has the opposite entries below and above the diagonal: aij = ¡aji for any values of i and j between 1 and n. This automatically means the digaonal entries must all be zero. Transpose To transpose a matrix, we reect it across the line given by the leading diagonal a11, a22 etc. In general the result is a di®erent shape to the original matrix: a11 a21 a11 a12 a13 > > A = A = 0 a12 a22 1 [A ]ij = A : µ a21 a22 a23 ¶ ji a13 a23 @ A > ² If A is m £ n then A is n £ m. > ² The transpose of a symmetric matrix is itself: A = A (recalling that only square matrices can be symmetric).
    [Show full text]
  • Triangular Factorization
    Chapter 1 Triangular Factorization This chapter deals with the factorization of arbitrary matrices into products of triangular matrices. Since the solution of a linear n n system can be easily obtained once the matrix is factored into the product× of triangular matrices, we will concentrate on the factorization of square matrices. Specifically, we will show that an arbitrary n n matrix A has the factorization P A = LU where P is an n n permutation matrix,× L is an n n unit lower triangular matrix, and U is an n ×n upper triangular matrix. In connection× with this factorization we will discuss pivoting,× i.e., row interchange, strategies. We will also explore circumstances for which A may be factored in the forms A = LU or A = LLT . Our results for a square system will be given for a matrix with real elements but can easily be generalized for complex matrices. The corresponding results for a general m n matrix will be accumulated in Section 1.4. In the general case an arbitrary m× n matrix A has the factorization P A = LU where P is an m m permutation× matrix, L is an m m unit lower triangular matrix, and U is an×m n matrix having row echelon structure.× × 1.1 Permutation matrices and Gauss transformations We begin by defining permutation matrices and examining the effect of premulti- plying or postmultiplying a given matrix by such matrices. We then define Gauss transformations and show how they can be used to introduce zeros into a vector. Definition 1.1 An m m permutation matrix is a matrix whose columns con- sist of a rearrangement of× the m unit vectors e(j), j = 1,...,m, in RI m, i.e., a rearrangement of the columns (or rows) of the m m identity matrix.
    [Show full text]
  • 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
    [Show full text]
  • 8.3 Positive Definite Matrices
    8.3. Positive Definite Matrices 433 Exercise 8.2.25 Show that every 2 2 orthog- [Hint: If a2 + b2 = 1, then a = cos θ and b = sinθ for × cos θ sinθ some angle θ.] onal matrix has the form − or sinθ cosθ cos θ sin θ Exercise 8.2.26 Use Theorem 8.2.5 to show that every for some angle θ. sinθ cosθ symmetric matrix is orthogonally diagonalizable. − 8.3 Positive Definite Matrices All the eigenvalues of any symmetric matrix are real; this section is about the case in which the eigenvalues are positive. These matrices, which arise whenever optimization (maximum and minimum) problems are encountered, have countless applications throughout science and engineering. They also arise in statistics (for example, in factor analysis used in the social sciences) and in geometry (see Section 8.9). We will encounter them again in Chapter 10 when describing all inner products in Rn. Definition 8.5 Positive Definite Matrices A square matrix is called positive definite if it is symmetric and all its eigenvalues λ are positive, that is λ > 0. Because these matrices are symmetric, the principal axes theorem plays a central role in the theory. Theorem 8.3.1 If A is positive definite, then it is invertible and det A > 0. Proof. If A is n n and the eigenvalues are λ1, λ2, ..., λn, then det A = λ1λ2 λn > 0 by the principal axes theorem (or× the corollary to Theorem 8.2.5). ··· If x is a column in Rn and A is any real n n matrix, we view the 1 1 matrix xT Ax as a real number.
    [Show full text]
  • Wronskian Solutions to the Kdv Equation Via B\" Acklund
    Wronskian solutions to the KdV equation via B¨acklund transformation Qi-fei Xuan∗, Mei-ying Ou, Da-jun Zhang† Department of Mathematics, Shanghai University, Shanghai 200444, P.R. China October 27, 2018 Abstract In the paper we discuss the B¨acklund transformation of the KdV equation between solitons and solitons, between negatons and negatons, between positons and positons, between rational solution and rational solution, and between complexitons and complexitons. We investigate the conditions that Wronskian entries satisfy for the bilinear B¨acklund transformation of the KdV equation. By choosing suitable Wronskian entries and the parameter in the bilinear B¨acklund transformation, we obtain transformations between many kinds of solutions. Keywords: the KdV equation, Wronskian solution, bilinear form, B¨acklund transformation 1 Introduction The Wronskian can be considered as a bridge connecting with many classical methods in soliton theory. This is not only because soliton solutions in Wronskian form can be obtained from the Darboux transformation[1], Sato theory[2, 3] and Wronskian technique[4]-[10], but also because the exponential polynomial for N-solitons derived from Hirota method[11, 12] and the matrix form given by the Inverse Scattering Transform[13, 14] can be transformed into a Wronskian by extracting some exponential factors. The special structure of a Wronskian contributes simple arXiv:0706.3487v1 [nlin.SI] 24 Jun 2007 forms of its derivatives, and this admits solution verification by direct substituting Wronskians into a bilinear soliton equation or a bilinear B¨acklund transformation(BT). This approach is re- ferred to as Wronskian technique[4]. In the approach a bilinear soliton equation is some algebraic identity provided that Wronskian entry vector satisfies some differential equation set which we call Wronskian condition.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]