PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY MOORE PENROSE INVERSE OF BIDIAGONAL MATRICES. I

Size: px
Start display at page:

Download "PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY MOORE PENROSE INVERSE OF BIDIAGONAL MATRICES. I"

Transcription

1 PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY Physical and Mathematical Sciences 205, 2, p. 20 M a t h e m a t i c s MOORE PENROSE INVERSE OF BIDIAGONAL MATRICES. I Yu. R. HAKOPIAN, S. S. ALEKSANYAN 2, Chair of Numerical Analysis and Mathematical Modeling YSU, Armenia 2 Chair of Mathematics and Mathematical Modeling RAU, Armenia In the present paper we deduce closed form expressions for the entries of the Moore Penrose inverse of a special type upper bidiagonal matrices. On the base of the formulae obtained, a finite algorithm with optimal order of computational complexity is constructed. MSC200: 5A09; 65F20. Keywords: generalized inverse, Moore Penrose inverse, bidiagonal matrix. Introduction. For a real m n matrix A, the Moore Penrose inverse A + is the unique n m matrix that satisfies the following four properties: AA + A = A, A + AA + = A +, (A + A T = A + A, (AA + T = AA + (see [], for example. If A is a square nonsingular matrix, then A + = A. Thus, the Moore Penrose inversion generalizes ordinary matrix inversion. The idea of matrix generalized inverse was first introduced in 920 by E. Moore [2], and was later rediscovered by R. Penrose [3]. The Moore Penrose inverses have numerous applications in least-squares problems, regressive analysis, linear programming and etc. For more information on the generalized inverses see [] and its extensive bibliography. In this work we consider the problem of the Moore Penrose inversion of square upper bidiagonal matrices A = d b d 2 b d b d n. ( A natural question arises: what caused an interest in pseudoinversion of such matrices? The reason is as follows. The most effective procedure of computing yuri.hakopian@ysu.am alexosarm@gmail.com

2 2 Proc. of the Yerevan State Univ., Phys. and Math. Sci., 205, 2, p. 20. the Moore Penrose inverse involves two main steps [4]. In the first step an original matrix by means of Householder transformations is reduced to an upper bidiagonal form. Thus, there arises the problem of pseudoinversion of bidiagonal matrices of the form (. In the second step an iterative procedure, which is known as Golub Reinsch algorithm is implemented [4]. The procedure, by means of Givens rotations generates a sequence of matrices which converge to a diagonal matrix. As a result, at a certain step of the process we get an approximation of the singular value decomposition of the original bidiagonal matrix, by means of which, using a well-known formula (see [, 4], for example, the Moore Penrose inverse of this matrix is found. In this paper we develop an approach, which allows to deduce formulae for the entries of the Moore Penrose inverse of upper bidiagonal matrices. Obtained closed form solution for the Moore Penrose inversion may be considered as alternative to sufficiently labour-consuming Golub Reinsch iterative procedure. Moreover, explicit expressions for the entries of the Moore Penrose inverse lead to a simple algorithm of their computation. From now on, we will assume that b,b 2,...,b 0. (2 Otherwise, if some of over-diagonal entries of the matrix A from ( are equal to zero, then the original problem is decomposed into several similar problems for bidiagonal matrices of lower order. Next, we assume that the bidiagonal matrix ( is singular, i.e. d d 2...d n = 0. We begin by considering the special case, where d,d 2,...,d 0, d n = 0. (3 This particular case forms the problem solution basis in general case, that is, for an arbitrary arrangement of one or more zeros on the main diagonal of the matrix (. The Idea of the Derivation of Pseudoinversion Formulae. An approach to derive the formulae is based upon the well-known equality A + = lim ε +0 (AT A + εi A T, (4 where I is an identity matrix, which holds true for any real m n matrix []. Proposed way to obtain the matrix A + for the matrix ( consists of the following. Finding first the inverse matrix (A T A + εi, the entries of the matrix (A T A + εi A T are computed and the character of their dependence on the parameter ε is revealed. Then, according to the equality (4, passing to the limit when ε +0, we arrive to a closed form expressions for the entries of the matrix A +. Mention that a similar approach was also used in the papers [5, 6]. Thus, the first problem that arise is the inversion of the matrix A T A + εi. For our bidiagonal matrix (, under the assumption d n = 0 (see (3, we have A T A + εi = d 2 + ε b d b d b 2 + d2 2 + ε b 2d b n 2 d n 2 b 2 n 2 + d2 + ε b d b d b 2 + ε. (5

3 Hakopian Yu. R., Aleksanian S. S. Moore Penrose Inverse of Bidiagonal Matrices. I. 3 To invert this matrix, let us take advantage of an algorithm developed in [7]. Consider a nonsingulaymmetric tridiagonal matrix c c 2 c 2 c 22 c 23 0 C = , (6 0 c n 2 c c n c n c nn where c ii = c i i 0, i = 2,3,...,n. Referring to [7], the matrix C = [x i j ] n n can be obtained by the following computational procedure. Algorithm 3d/inv (C C. Compute the quantities f i (i = 2,3,...,n, g i (i = 2,3,...,n and h i (i =,2,...,n : f i = c ii, g i = c ii+, h i = c ii. (7 c ii c ii c ii+ 2. Compute recursively the quantities: µ n =, µ = f n, µ i = f i+ µ i+ g i+ µ i+2, i = n 2,n 3,...,. 3. Compute recursively the quantities: ν =, ν 2 = h, ν i = h i ν i g i ν i 2, i = 3,4,...,n. 4. Compute the quantity t = (c µ + c 2 µ 2. (0 Note: if C is nonsingular matrix then c µ + c 2 µ 2 0 [7]. 5. The entries of the lower triangular part of the matrix C are computed by: x i j = µ i ν j t, i = j, j +,...,n; j =,2,...,n. ( 6. The entries of the upper triangular part of the matrix C are computed by: End. x i j = µ j ν i t, i =,2,..., j ; j = 2,3,...,n. (2 Consider as the matrix C our tridiagonal matrix A T A + εi. Comparing the records of matrices (5 and (6, we have (8 (9 c ii = b 2 i + d 2 i + ε, i =,2,...,n (3 (for the purpose of unification the record of the formulae, we set b 0 = 0 and c ii+ = b i d i, i =,2,...,n ; c ii = b i d i, i = 2,3,...,n. (4 The Entries of the Inverse Matrix (A T A + εi. Let us carry out a more detailed study of the quantities successively computed in the algorithm 3d/inv. More precisely, we are interested in revealing the nature of dependence of these quantities on the parameter ε.

4 4 Proc. of the Yerevan State Univ., Phys. and Math. Sci., 205, 2, p. 20. Consider first the quantities f i, g i, h i, which were introduced in (7. Using the expressions (3 and (4, we get f i = f i +α i ε, i = 2,3,...,n, where f i = b2 i + d2 i, α i = ; b i d i b i d i (5 g i = b id i, i = 2,3,...,n ; b i d i (6 h i = hi +β i ε, i =,2,...,n, where hi= b2 i + d2 i, β i =. b i d i b i d i (7 Next, go to the quantities µ i and ν i recursively defined in (8 and (9 respectively. L e m m a. The quantities µ i may be represented as µ i = µ i +γ i ε + O(ε 2, i =,2,...,n, (8 where the quantities µ i and γ i satisfy the following recurrence relations: µ n =, µ = f n, µ i = f i+ µi+ g i+ µi+2, i = n 2,n 3,...,, and γ n = 0, γ = α n, γ i = f i+ γ i+ g i+ γ i+2 α i+ µi+, i = n 2,n 3,...,. (9 (20 P r o o f. Since µ n =, then in (8 for i = n we set µ n =, γ n = 0. Further, µ = f n (see (8. According to the expressions (5, we have f n = f n +α n ε. Therefore, in the representation (8 for i = n we set µ = f n, γ = α n. Required representations for the indices in the range i n 2 can be readily derived by induction from the relations (8, using expressions (5. Indeed, having µ i = f i+ µ i+ g i+ µ i+2 = = ( f i+ +α i+ ε( µ i+ +γ i+ ε + O(ε 2 g i+ ( µ i+2 +γ i+2 ε + O(ε 2 = = ( f i+µi+ g i+ µi+2 + ( f i+ γ i+ g i+ γ i+2 α i+ µi+ ε + O(ε 2, we get (8, as well as the recurrence relations (9 and (20. At the same time, the quantities µ i computed by the recursion (9 may be represented in closed form. Let us introduce the following notation: b s, d s s =,2,...,n. (2 Additionally, we set r 0 = r n =. L e m m a 2. The quantities µ i may be written as µ i = ( n i s=i, i =,2,...,n. (22

5 Hakopian Yu. R., Aleksanian S. S. Moore Penrose Inverse of Bidiagonal Matrices. I. 5 P r o o f. Firstly, the value µ n = conforms to the record (22. Then, in accordance with (5, µ = f n = b d = r. Further reasonings are carried out by induction. Taking into account the expressions (5 and (6, we have µ i = b2 i + d2 i+ ( n i b i d i s=i+ b i+d i+ b i d i ( n i 2 = ( n i ( b 2 r i + di+ 2 s r i+ b i+d i+ = ( s=i+2 b i d i b i d i which completes the proof of the Lemma. The next assertion is a simple consequence of the equality (22. C o r o l l a r y. The following relation holds s=i+2 n i s=i µ i = r i µi+, i =,2,...,n. (23 A representation similar to (8 takes place also for the quantities ν i. It can be obtained using the relations (9 and the expressions (7. L e m m a 3. The quantities ν i may be represented as ν i = ν i +δ i ε + O(ε 2, i =,2,...,n, (24 r i, where the quantities ν i and δ i satisfy the following recurrence relations: and ν =, ν 2 = h, ν i = hi ν i νi 2, i = 3,4,...,n, g i δ = 0, δ 2 = β, δ i = hi δ i g i δ i 2 β i νi, i = 3,4,...,n. (25 (26 The quantities ν i may be represented in closed form as well. L e m m a 4. The quantities ν i may be written by i i+ ν i = (, i =,2,...,n. (27 The proof of the Lemma is similar to the one of Lemma 2, using relations (25 and expressions (6, (7. As a simple consequence of the equality (27 we obtain the following statement. C o r o l l a r y 2. The following relation holds ν i+ = r i νi, i =,2,...,n. (28

6 6 Proc. of the Yerevan State Univ., Phys. and Math. Sci., 205, 2, p. 20. Our next task is to get the expression for the quantity t from (0, depending on the parameter ε. Since c = d 2 +ε, c 2 = b d (compare (5 and (6, then taking into account the representations (8 for the quantities µ i, we get t = ((d 2 + ε( µ +γ ε + O(ε 2 + b d ( µ 2 +γ 2 ε + O(ε 2 = = (d 2 ( µ +r µ2 + ( µ +d (γ d + γ 2 b ε + O(ε 2. By virtue of the relation (23, µ +r µ2 = 0. Therefore, t = (( µ +d (γ d + γ 2 b ε + O(ε 2. (29 Finally, having the representations for the quantities µ i, ν i and t, we can obtain the entries of the inverse matrix (A T A + εi = [x i j ] n n and reveal their dependence on the parameter ε. It may be done by the equalities ( and (2, given in the algorithm 3d/inv. We leave it to the next section. Formulae for the Entries of the Matrix A +. Let (A T A + εi A T Y (ε = [y i j (ε] n n, A + = [a i j ] n n. (30 Since A + = lim Y (ε, then ε +0 lim ε +0 y i j(ε, i, j =,2,...,n. (3 The entries of the last column of the matrix A T are equal to zero (remind that d n = 0. Hence, y in (ε = 0, i =,2,...,n, and by this very fact a in = 0, i =,2,...,n. (32 From (30 it is seen that the entries of the matrix Y (ε, for indices i =,2,...,n and j =,2,...,n, are calculated by the rule y i j (ε = x i j d j + x i j+ b j. (33 The formulae ( and (2 by which we calculate lower and upper triangular parts of the matrix (A T A + εi respectively, are somewhat different. Therefore, for a fixed index j in the range j n we consideeparately two cases: i =,2,..., j and i = j +, j + 2,...,n. Case i =,2,..., j. Taking advantage of the expression (2 for the entries x i j, from (33 we have y i j (ε = tν i (µ j d j + µ j+ b j. (34 Then, using the representations (8 for the quantities µ i, we can write that µ j d j + µ j+ b j = ( µ j +γ j ε + O(ε 2 d j + ( µ j+ +γ j+ ε + O(ε 2 b j = = ( µ j d j + µ j+ b j + (γ j d j + γ j+ b j ε + O(ε 2. As follows from the relation (23, µ j d j + µ j+ b j = d j ( µ j +r j µ j+ = 0. Hence, µ j d j + µ j+ b j = (γ j d j + γ j+ b j ε + O(ε 2. (35

7 Hakopian Yu. R., Aleksanian S. S. Moore Penrose Inverse of Bidiagonal Matrices. I. 7 Substituting the expression (35 as well as the representations (24 and (29 of the quantities ν i and t respectively, into the right hand side of the equality (34 yields y i j (ε = ν i (γ j d j + γ j+ b j + O(ε. µ +d (γ d + γ 2 b + O(ε From here, by taking limit in the equality as ε +0 (see (3, we find that If we introduce the notation ν i (γ j d j + γ j+ b j, i =,2,..., j. (36 µ +d (γ d + γ 2 b u j γ j d j + γ j+ b j, j =,2,...,n, (37 the entries a i j from (36 may be written as follows: ν i u j, i =,2,..., j. (38 µ +d u Using expressions (5, (6 and (20, one may show that the quantities u j, defined in (37, satisfy the following relations: µ j+ u =, u j = d j+u j+ +, j = n 2,n 3,...,. (39 b b j Moreover, it appears that the quantities u j may be written as u j = ( n j ( b j, j =,2,...,n. (40 n j ( n k s= j+ s=n k+ It can be proved by substituting the expression (40 into the relations (39. Finally, let us replace the expression (40 for the quantities u j as well as the expressions (22 and (27 for µ i and ν i respectively into (38. As a result, we obtain the following closed form expression for the entries of the upper triangular part of the matrix A + : ( i n j i+ j b j n ( n k s= j+ Case i = j +, j + 2,...,n. ( ( n k ( s=n k+ s=n k+, i =,2,..., j. (4 Using the expressions ( for the entries x i j, from (33 we have In accordance with representations (24, we write y i j (ε = tµ i (ν j d j + ν j+ b j. (42 ν j d j + ν j+ b j = ( ν j +δ j ε + O(ε 2 d j + ( ν j+ +δ j+ ε + O(ε 2 b j = = ( ν j d j + ν j+ b j + (δ j d j + δ j+ b j ε + O(ε 2.

8 8 Proc. of the Yerevan State Univ., Phys. and Math. Sci., 205, 2, p. 20. As follows from the relation (28, ν j d j + ν j+ b j = d j ( ν j +r j ν j+ = 0. Hence, ν j d j + ν j+ b j = (δ j d j + δ j+ b j ε + O(ε 2. (43 If we substitute expression (43 as well as the representations (8 and (29 of the quantities µ i and t respectively into the right hand side of the equality (42, we get µ i (δ j d j + δ j+ b j + O(ε y i j (ε =. µ +d (γ d + γ 2 b + O(ε By taking limit in the last equality as ε +0, we obtaine µ i (δ j d j + δ j+ b j, i = j +,...,n. (44 µ +d (γ d + γ 2 b Similarly to the previous case, we introduce the notation w j δ j d j + δ j+ b j, j =,2,...,n. (45 Then the entries a i j from (44 may be rewritten by µ i w j, i = j +, j + 2,...,n. (46 µ +d u Having expressions (6, (7 and (26, we find that defined in (45 quantities w j satisfy the following relations: w = d, w j = b j w j + ν j d j, j = 2,3,...,n. (47 It turns out, however, that the quantities w j may be written by w j = ( j ( k ( j d j j s=k, j =,2,...,n. (48 It may readily make sure by substituting the expression (48 into the relations (47. Now replace the expression (48 for the quantity w j and the expression (22 of µ i into the equality (46. Resulting formula for the entries of lower triangular part of the matrix A + is of the following type: ( ( k ( i+ j+ n d j s=i ( n k j ( s=n k+ ( j s=k, i = j +,...,n. (49 Thus, in (32, (4 and (49 we have got the entries of the matrix A +, expressed through the entries of the original bidiagonal matrix A. Summarizing the considerations of the section, we arrive at the following main statement. T h e o r e m. Let A be a bidiagonal matrix of the form (, whose entries meet the conditions (2 and (3. Then the entries of the Moore Penrose inverse matrix A + = [a i j ] n n are as follows:

9 Hakopian Yu. R., Aleksanian S. S. Moore Penrose Inverse of Bidiagonal Matrices. I. 9 a for the indices j =,2,...,n : n j ( n k ( ( i+ j s= j+ n ( n k ( b j ( i i+ j+ n d j s=i j ( n k ( ( k s=n k+ s=n k+ ( j s=k s=n k+, i =,2,..., j ; (50, i = j +, j + 2,...,n, (5 where = b s /d s, s =,2,...,n and r 0 = r n = ; b for the index j = n: a in = 0, i =,2,...,n. (52 Below is an example to illustrate the Theorem. E x a m p l e. Consider the n n matrix A = According to the Eqs. (50 and (5, the entries of the first n columns of the matrix A + are as follows. For the indices j =,2,...,n :. ( i+ j ( j/n, i =,2,..., j, ( i+ j+ j/n, i = j +, j + 2,...,n. In the next section we discuss the issues connected with the practical computation of the matrix A +. An Algorithm to Compute the Matrix A +. Obtained in the previous sections intermediate formulae and relations allow us to propose the following fairly simple algorithm for computing the entries of the matrix A +. Algorithm 2d/pinv/special (A A +. Compute the quantities (see (2: = b s /d s, s =,2,...,n. 2. Compute the quantities µ i (see (9, (23: µ n = ; µ i = r i µi+, i = n,n 2,...,. 3. Compute the quantities ν i (see (25, (28: ν = ; 4. Compute the quantities u j (see (39: ν i+ = ν i /r i, i =,2,...,n. u = /b ;u j = (d j+ u j+ + µ j+ /b j, j = n 2,n 3,...,.

10 20 Proc. of the Yerevan State Univ., Phys. and Math. Sci., 205, 2, p Compute the quantities w j (see (47: w = /d ; w j = (b j w j + ν j /d j, j = 2,3,...,n. 6. Set a in = 0, i =,2,...,n (see ( Compute the entries of upper triangular part of the matrix A + (see (38: ν i u j /( µ +d u, i =,2,..., j ; j =,2,...,n. 8. Compute the entries of lower triangular part of the matrix A + (see (46: µ i w j /( µ +d u, i = j +, j + 2,...,n; j =,2,...,n. End. Direct calculations show that the numerical implementation of the algorithm 2d/pinv/special requires O(n 2 arithmetical operations. By this very fact, the algorithm may be considered as an optimal one. Concluding Remarks. In this paper we have deduced closed form expressions, as well as the numerical algorithm, to compute the entries of the Moore Penrose inverse of bidiagonal matrices (, under assumptions (2 and (3. On the base of obtained resultes in a subsequent paper we will consider bidiagonal matrices of the form ( with any arrangement of one or more zeros on the main diagonal. Received R E F E R E N C E S. Ben-Israel A., Greville T.N.E. Generalized Inverses. Theory and Applications (2nd ed.. New York: Springer, Moore E. On the Reciprocal of the General Algebraic Matrix. // Bull. Amer. Math. Soc., 920, v. 26, p Penrose R. A Generalized Inverse for Matrices. // Proc. Cambridge Philos. Soc., 955, v. 5, p Golub G.H., van Loan Ch.F. Matrix Computations (3rd ed.. The Johns Hopkins University Press, Martirosyan V.A. An Algorithm for Pseudoinversion of Bidiagonal Matrices. // Vestnik RAU. Fiziko-Matematicheskie i Estestvennye Nauki, 2008,, p (in Russian. 6. Hakopian Yu.R., Martirosyan V.A. About Parallelization of the Algorithm for Pseudoinversion of Matrices. // Proceedings of State Engineering University of Armenia. Modelling, Optimization, Control, 2009, col. 2, v., p (in Russian. 7. Lewis J.W. Invertion of Tridiagonal Matrices. // Numer. Math., 982, v. 38, p

A Note on Solutions of the Matrix Equation AXB = C

A Note on Solutions of the Matrix Equation AXB = C SCIENTIFIC PUBLICATIONS OF THE STATE UNIVERSITY OF NOVI PAZAR SER A: APPL MATH INFORM AND MECH vol 6, 1 (214), 45-55 A Note on Solutions of the Matrix Equation AXB C I V Jovović, B J Malešević Abstract:

More information

a Λ q 1. Introduction

a Λ q 1. Introduction International Journal of Pure and Applied Mathematics Volume 9 No 26, 959-97 ISSN: -88 (printed version); ISSN: -95 (on-line version) url: http://wwwijpameu doi: 272/ijpamv9i7 PAijpameu EXPLICI MOORE-PENROSE

More information

Moore-Penrose s inverse and solutions of linear systems

Moore-Penrose s inverse and solutions of linear systems Available online at www.worldscientificnews.com WSN 101 (2018) 246-252 EISSN 2392-2192 SHORT COMMUNICATION Moore-Penrose s inverse and solutions of linear systems J. López-Bonilla*, R. López-Vázquez, S.

More information

A note on solutions of linear systems

A note on solutions of linear systems A note on solutions of linear systems Branko Malešević a, Ivana Jovović a, Milica Makragić a, Biljana Radičić b arxiv:134789v2 mathra 21 Jul 213 Abstract a Faculty of Electrical Engineering, University

More information

An iterative method for the skew-symmetric solution and the optimal approximate solution of the matrix equation AXB =C

An iterative method for the skew-symmetric solution and the optimal approximate solution of the matrix equation AXB =C Journal of Computational and Applied Mathematics 1 008) 31 44 www.elsevier.com/locate/cam An iterative method for the skew-symmetric solution and the optimal approximate solution of the matrix equation

More information

Re-nnd solutions of the matrix equation AXB = C

Re-nnd solutions of the matrix equation AXB = C Re-nnd solutions of the matrix equation AXB = C Dragana S. Cvetković-Ilić Abstract In this article we consider Re-nnd solutions of the equation AXB = C with respect to X, where A, B, C are given matrices.

More information

Proceedings of the Third International DERIV/TI-92 Conference

Proceedings of the Third International DERIV/TI-92 Conference Proceedings of the Third International DERIV/TI-9 Conference The Use of a Computer Algebra System in Modern Matrix Algebra Karsten Schmidt Faculty of Economics and Management Science, University of Applied

More information

Numerical Linear Algebra Homework Assignment - Week 2

Numerical Linear Algebra Homework Assignment - Week 2 Numerical Linear Algebra Homework Assignment - Week 2 Đoàn Trần Nguyên Tùng Student ID: 1411352 8th October 2016 Exercise 2.1: Show that if a matrix A is both triangular and unitary, then it is diagonal.

More information

Weaker assumptions for convergence of extended block Kaczmarz and Jacobi projection algorithms

Weaker assumptions for convergence of extended block Kaczmarz and Jacobi projection algorithms DOI: 10.1515/auom-2017-0004 An. Şt. Univ. Ovidius Constanţa Vol. 25(1),2017, 49 60 Weaker assumptions for convergence of extended block Kaczmarz and Jacobi projection algorithms Doina Carp, Ioana Pomparău,

More information

Data Analysis and Manifold Learning Lecture 2: Properties of Symmetric Matrices and Examples

Data Analysis and Manifold Learning Lecture 2: Properties of Symmetric Matrices and Examples Data Analysis and Manifold Learning Lecture 2: Properties of Symmetric Matrices and Examples Radu Horaud INRIA Grenoble Rhone-Alpes, France Radu.Horaud@inrialpes.fr http://perception.inrialpes.fr/ Outline

More information

Preliminary/Qualifying Exam in Numerical Analysis (Math 502a) Spring 2012

Preliminary/Qualifying Exam in Numerical Analysis (Math 502a) Spring 2012 Instructions Preliminary/Qualifying Exam in Numerical Analysis (Math 502a) Spring 2012 The exam consists of four problems, each having multiple parts. You should attempt to solve all four problems. 1.

More information

Lecture 5 Singular value decomposition

Lecture 5 Singular value decomposition Lecture 5 Singular value decomposition Weinan E 1,2 and Tiejun Li 2 1 Department of Mathematics, Princeton University, weinan@princeton.edu 2 School of Mathematical Sciences, Peking University, tieli@pku.edu.cn

More information

Dragan S. Djordjević. 1. Introduction and preliminaries

Dragan S. Djordjević. 1. Introduction and preliminaries PRODUCTS OF EP OPERATORS ON HILBERT SPACES Dragan S. Djordjević Abstract. A Hilbert space operator A is called the EP operator, if the range of A is equal with the range of its adjoint A. In this article

More information

An Alternative Proof of the Greville Formula

An Alternative Proof of the Greville Formula JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS: Vol. 94, No. 1, pp. 23-28, JULY 1997 An Alternative Proof of the Greville Formula F. E. UDWADIA1 AND R. E. KALABA2 Abstract. A simple proof of the Greville

More information

Interval solutions for interval algebraic equations

Interval solutions for interval algebraic equations Mathematics and Computers in Simulation 66 (2004) 207 217 Interval solutions for interval algebraic equations B.T. Polyak, S.A. Nazin Institute of Control Sciences, Russian Academy of Sciences, 65 Profsoyuznaya

More information

Some results on the reverse order law in rings with involution

Some results on the reverse order law in rings with involution Some results on the reverse order law in rings with involution Dijana Mosić and Dragan S. Djordjević Abstract We investigate some necessary and sufficient conditions for the hybrid reverse order law (ab)

More information

Moore-Penrose-invertible normal and Hermitian elements in rings

Moore-Penrose-invertible normal and Hermitian elements in rings Moore-Penrose-invertible normal and Hermitian elements in rings Dijana Mosić and Dragan S. Djordjević Abstract In this paper we present several new characterizations of normal and Hermitian elements in

More information

We first repeat some well known facts about condition numbers for normwise and componentwise perturbations. Consider the matrix

We first repeat some well known facts about condition numbers for normwise and componentwise perturbations. Consider the matrix BIT 39(1), pp. 143 151, 1999 ILL-CONDITIONEDNESS NEEDS NOT BE COMPONENTWISE NEAR TO ILL-POSEDNESS FOR LEAST SQUARES PROBLEMS SIEGFRIED M. RUMP Abstract. The condition number of a problem measures the sensitivity

More information

arxiv: v4 [quant-ph] 8 Mar 2013

arxiv: v4 [quant-ph] 8 Mar 2013 Decomposition of unitary matrices and quantum gates Chi-Kwong Li, Rebecca Roberts Department of Mathematics, College of William and Mary, Williamsburg, VA 2387, USA. (E-mail: ckli@math.wm.edu, rlroberts@email.wm.edu)

More information

Properties of Matrices and Operations on Matrices

Properties of Matrices and Operations on Matrices Properties of Matrices and Operations on Matrices A common data structure for statistical analysis is a rectangular array or matris. Rows represent individual observational units, or just observations,

More information

ADDITIVE RESULTS FOR THE GENERALIZED DRAZIN INVERSE

ADDITIVE RESULTS FOR THE GENERALIZED DRAZIN INVERSE ADDITIVE RESULTS FOR THE GENERALIZED DRAZIN INVERSE Dragan S Djordjević and Yimin Wei Abstract Additive perturbation results for the generalized Drazin inverse of Banach space operators are presented Precisely

More information

SOLVING FUZZY LINEAR SYSTEMS BY USING THE SCHUR COMPLEMENT WHEN COEFFICIENT MATRIX IS AN M-MATRIX

SOLVING FUZZY LINEAR SYSTEMS BY USING THE SCHUR COMPLEMENT WHEN COEFFICIENT MATRIX IS AN M-MATRIX Iranian Journal of Fuzzy Systems Vol 5, No 3, 2008 pp 15-29 15 SOLVING FUZZY LINEAR SYSTEMS BY USING THE SCHUR COMPLEMENT WHEN COEFFICIENT MATRIX IS AN M-MATRIX M S HASHEMI, M K MIRNIA AND S SHAHMORAD

More information

Krylov Subspace Methods that Are Based on the Minimization of the Residual

Krylov Subspace Methods that Are Based on the Minimization of the Residual Chapter 5 Krylov Subspace Methods that Are Based on the Minimization of the Residual Remark 51 Goal he goal of these methods consists in determining x k x 0 +K k r 0,A such that the corresponding Euclidean

More information

ON THE TWO-LEVEL PRECONDITIONING IN LEAST SQUARES METHOD

ON THE TWO-LEVEL PRECONDITIONING IN LEAST SQUARES METHOD PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY Physical and Matheatical Sciences 04,, p. 7 5 ON THE TWO-LEVEL PRECONDITIONING IN LEAST SQUARES METHOD M a t h e a t i c s Yu. A. HAKOPIAN, R. Z. HOVHANNISYAN

More information

Stirling Numbers of the 1st Kind

Stirling Numbers of the 1st Kind Daniel Reiss, Colebrook Jackson, Brad Dallas Western Washington University November 28, 2012 Introduction The set of all permutations of a set N is denoted S(N), while the set of all permutations of {1,

More information

arxiv: v1 [math.ra] 14 Apr 2018

arxiv: v1 [math.ra] 14 Apr 2018 Three it representations of the core-ep inverse Mengmeng Zhou a, Jianlong Chen b,, Tingting Li c, Dingguo Wang d arxiv:180.006v1 [math.ra] 1 Apr 018 a School of Mathematics, Southeast University, Nanjing,

More information

Discrete mathematical models in the analysis of splitting iterative methods for linear systems

Discrete mathematical models in the analysis of splitting iterative methods for linear systems Computers and Mathematics with Applications 56 2008) 727 732 wwwelseviercom/locate/camwa Discrete mathematical models in the analysis of splitting iterative methods for linear systems Begoña Cantó, Carmen

More information

Applied Mathematics Letters. Comparison theorems for a subclass of proper splittings of matrices

Applied Mathematics Letters. Comparison theorems for a subclass of proper splittings of matrices Applied Mathematics Letters 25 (202) 2339 2343 Contents lists available at SciVerse ScienceDirect Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml Comparison theorems for a subclass

More information

Diagonal and Monomial Solutions of the Matrix Equation AXB = C

Diagonal and Monomial Solutions of the Matrix Equation AXB = C Iranian Journal of Mathematical Sciences and Informatics Vol. 9, No. 1 (2014), pp 31-42 Diagonal and Monomial Solutions of the Matrix Equation AXB = C Massoud Aman Department of Mathematics, Faculty of

More information

On V-orthogonal projectors associated with a semi-norm

On V-orthogonal projectors associated with a semi-norm On V-orthogonal projectors associated with a semi-norm Short Title: V-orthogonal projectors Yongge Tian a, Yoshio Takane b a School of Economics, Shanghai University of Finance and Economics, Shanghai

More information

Block Bidiagonal Decomposition and Least Squares Problems

Block Bidiagonal Decomposition and Least Squares Problems Block Bidiagonal Decomposition and Least Squares Problems Åke Björck Department of Mathematics Linköping University Perspectives in Numerical Analysis, Helsinki, May 27 29, 2008 Outline Bidiagonal Decomposition

More information

Orthogonalization and least squares methods

Orthogonalization and least squares methods Chapter 3 Orthogonalization and least squares methods 31 QR-factorization (QR-decomposition) 311 Householder transformation Definition 311 A complex m n-matrix R = [r ij is called an upper (lower) triangular

More information

ETNA Kent State University

ETNA Kent State University C 8 Electronic Transactions on Numerical Analysis. Volume 17, pp. 76-2, 2004. Copyright 2004,. ISSN 1068-613. etnamcs.kent.edu STRONG RANK REVEALING CHOLESKY FACTORIZATION M. GU AND L. MIRANIAN Abstract.

More information

Moore Penrose inverses and commuting elements of C -algebras

Moore Penrose inverses and commuting elements of C -algebras Moore Penrose inverses and commuting elements of C -algebras Julio Benítez Abstract Let a be an element of a C -algebra A satisfying aa = a a, where a is the Moore Penrose inverse of a and let b A. We

More information

A revisit to a reverse-order law for generalized inverses of a matrix product and its variations

A revisit to a reverse-order law for generalized inverses of a matrix product and its variations A revisit to a reverse-order law for generalized inverses of a matrix product and its variations Yongge Tian CEMA, Central University of Finance and Economics, Beijing 100081, China Abstract. For a pair

More information

Solution of Linear Equations

Solution of Linear Equations Solution of Linear Equations (Com S 477/577 Notes) Yan-Bin Jia Sep 7, 07 We have discussed general methods for solving arbitrary equations, and looked at the special class of polynomial equations A subclass

More information

The DMP Inverse for Rectangular Matrices

The DMP Inverse for Rectangular Matrices Filomat 31:19 (2017, 6015 6019 https://doi.org/10.2298/fil1719015m Published by Faculty of Sciences Mathematics, University of Niš, Serbia Available at: http://.pmf.ni.ac.rs/filomat The DMP Inverse for

More information

A fast randomized algorithm for overdetermined linear least-squares regression

A fast randomized algorithm for overdetermined linear least-squares regression A fast randomized algorithm for overdetermined linear least-squares regression Vladimir Rokhlin and Mark Tygert Technical Report YALEU/DCS/TR-1403 April 28, 2008 Abstract We introduce a randomized algorithm

More information

Multiplicative Perturbation Bounds of the Group Inverse and Oblique Projection

Multiplicative Perturbation Bounds of the Group Inverse and Oblique Projection Filomat 30: 06, 37 375 DOI 0.98/FIL67M Published by Faculty of Sciences Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Multiplicative Perturbation Bounds of the Group

More information

ON ORTHOGONAL REDUCTION TO HESSENBERG FORM WITH SMALL BANDWIDTH

ON ORTHOGONAL REDUCTION TO HESSENBERG FORM WITH SMALL BANDWIDTH ON ORTHOGONAL REDUCTION TO HESSENBERG FORM WITH SMALL BANDWIDTH V. FABER, J. LIESEN, AND P. TICHÝ Abstract. Numerous algorithms in numerical linear algebra are based on the reduction of a given matrix

More information

Math 504 (Fall 2011) 1. (*) Consider the matrices

Math 504 (Fall 2011) 1. (*) Consider the matrices Math 504 (Fall 2011) Instructor: Emre Mengi Study Guide for Weeks 11-14 This homework concerns the following topics. Basic definitions and facts about eigenvalues and eigenvectors (Trefethen&Bau, Lecture

More information

Matrix & Linear Algebra

Matrix & Linear Algebra Matrix & Linear Algebra Jamie Monogan University of Georgia For more information: http://monogan.myweb.uga.edu/teaching/mm/ Jamie Monogan (UGA) Matrix & Linear Algebra 1 / 84 Vectors Vectors Vector: A

More information

EP elements in rings

EP elements in rings EP elements in rings Dijana Mosić, Dragan S. Djordjević, J. J. Koliha Abstract In this paper we present a number of new characterizations of EP elements in rings with involution in purely algebraic terms,

More information

Math 405: Numerical Methods for Differential Equations 2016 W1 Topics 10: Matrix Eigenvalues and the Symmetric QR Algorithm

Math 405: Numerical Methods for Differential Equations 2016 W1 Topics 10: Matrix Eigenvalues and the Symmetric QR Algorithm Math 405: Numerical Methods for Differential Equations 2016 W1 Topics 10: Matrix Eigenvalues and the Symmetric QR Algorithm References: Trefethen & Bau textbook Eigenvalue problem: given a matrix A, find

More information

Explicit Expressions for Free Components of. Sums of the Same Powers

Explicit Expressions for Free Components of. Sums of the Same Powers Applied Mathematical Sciences, Vol., 27, no. 53, 2639-2645 HIKARI Ltd, www.m-hikari.com https://doi.org/.2988/ams.27.79276 Explicit Expressions for Free Components of Sums of the Same Powers Alexander

More information

~ g-inverses are indeed an integral part of linear algebra and should be treated as such even at an elementary level.

~ g-inverses are indeed an integral part of linear algebra and should be treated as such even at an elementary level. Existence of Generalized Inverse: Ten Proofs and Some Remarks R B Bapat Introduction The theory of g-inverses has seen a substantial growth over the past few decades. It is an area of great theoretical

More information

Recurrence Relations and Recursion: MATH 180

Recurrence Relations and Recursion: MATH 180 Recurrence Relations and Recursion: MATH 180 1: Recursively Defined Sequences Example 1: The sequence a 1,a 2,a 3,... can be defined recursively as follows: (1) For all integers k 2, a k = a k 1 + 1 (2)

More information

On the Relative Gain Array (RGA) with Singular and Rectangular Matrices

On the Relative Gain Array (RGA) with Singular and Rectangular Matrices On the Relative Gain Array (RGA) with Singular and Rectangular Matrices Jeffrey Uhlmann University of Missouri-Columbia 201 Naka Hall, Columbia, MO 65211 5738842129, uhlmannj@missouriedu arxiv:180510312v2

More information

APPLICATIONS OF THE HYPER-POWER METHOD FOR COMPUTING MATRIX PRODUCTS

APPLICATIONS OF THE HYPER-POWER METHOD FOR COMPUTING MATRIX PRODUCTS Univ. Beograd. Publ. Eletrotehn. Fa. Ser. Mat. 15 (2004), 13 25. Available electronically at http: //matematia.etf.bg.ac.yu APPLICATIONS OF THE HYPER-POWER METHOD FOR COMPUTING MATRIX PRODUCTS Predrag

More information

Linear Systems of n equations for n unknowns

Linear Systems of n equations for n unknowns Linear Systems of n equations for n unknowns In many application problems we want to find n unknowns, and we have n linear equations Example: Find x,x,x such that the following three equations hold: x

More information

arxiv: v2 [quant-ph] 5 Dec 2013

arxiv: v2 [quant-ph] 5 Dec 2013 Decomposition of quantum gates Chi Kwong Li and Diane Christine Pelejo Department of Mathematics, College of William and Mary, Williamsburg, VA 23187, USA E-mail: ckli@math.wm.edu, dcpelejo@gmail.com Abstract

More information

Linear Algebra Section 2.6 : LU Decomposition Section 2.7 : Permutations and transposes Wednesday, February 13th Math 301 Week #4

Linear Algebra Section 2.6 : LU Decomposition Section 2.7 : Permutations and transposes Wednesday, February 13th Math 301 Week #4 Linear Algebra Section. : LU Decomposition Section. : Permutations and transposes Wednesday, February 1th Math 01 Week # 1 The LU Decomposition We learned last time that we can factor a invertible matrix

More information

MAT 1332: CALCULUS FOR LIFE SCIENCES. Contents. 1. Review: Linear Algebra II Vectors and matrices Definition. 1.2.

MAT 1332: CALCULUS FOR LIFE SCIENCES. Contents. 1. Review: Linear Algebra II Vectors and matrices Definition. 1.2. MAT 1332: CALCULUS FOR LIFE SCIENCES JING LI Contents 1 Review: Linear Algebra II Vectors and matrices 1 11 Definition 1 12 Operations 1 2 Linear Algebra III Inverses and Determinants 1 21 Inverse Matrices

More information

Moore [1, 2] introduced the concept (for matrices) of the pseudoinverse in. Penrose [3, 4] introduced independently, and much later, the identical

Moore [1, 2] introduced the concept (for matrices) of the pseudoinverse in. Penrose [3, 4] introduced independently, and much later, the identical J Soc INDUST APPL MATH Vol 11, No 2, June, 1963 Printed in USA A NOTE ON PSEUDOINVERSES C A DESOER AND B H WHALEN 1 Introduction When an operator, A, is invertible it is possible to solve the equation

More information

Bulletin of the. Iranian Mathematical Society

Bulletin of the. Iranian Mathematical Society ISSN 1017-060X (Print ISSN 1735-8515 (Online Bulletin of the Iranian Mathematical Society Vol 42 (2016, No 1, pp 53 60 Title The reverse order law for Moore-Penrose inverses of operators on Hilbert C*-modules

More information

arxiv: v1 [math.ra] 21 Jul 2013

arxiv: v1 [math.ra] 21 Jul 2013 Projections and Idempotents in -reducing Rings Involving the Moore-Penrose Inverse arxiv:1307.5528v1 [math.ra] 21 Jul 2013 Xiaoxiang Zhang, Shuangshuang Zhang, Jianlong Chen, Long Wang Department of Mathematics,

More information

Research Article A Generalization of a Class of Matrices: Analytic Inverse and Determinant

Research Article A Generalization of a Class of Matrices: Analytic Inverse and Determinant Advances in Numerical Analysis Volume 2011, Article ID 593548, 6 pages doi:10.1155/2011/593548 Research Article A Generalization of a Class of Matrices: Analytic Inverse and Determinant F. N. Valvi Department

More information

u n 2 4 u n 36 u n 1, n 1.

u n 2 4 u n 36 u n 1, n 1. Exercise 1 Let (u n ) be the sequence defined by Set v n = u n 1 x+ u n and f (x) = 4 x. 1. Solve the equations f (x) = 1 and f (x) =. u 0 = 0, n Z +, u n+1 = u n + 4 u n.. Prove that if u n < 1, then

More information

A New Algorithm for Solving Cross Coupled Algebraic Riccati Equations of Singularly Perturbed Nash Games

A New Algorithm for Solving Cross Coupled Algebraic Riccati Equations of Singularly Perturbed Nash Games A New Algorithm for Solving Cross Coupled Algebraic Riccati Equations of Singularly Perturbed Nash Games Hiroaki Mukaidani Hua Xu and Koichi Mizukami Faculty of Information Sciences Hiroshima City University

More information

Linear Algebra Methods for Data Mining

Linear Algebra Methods for Data Mining Linear Algebra Methods for Data Mining Saara Hyvönen, Saara.Hyvonen@cs.helsinki.fi Spring 2007 The Singular Value Decomposition (SVD) continued Linear Algebra Methods for Data Mining, Spring 2007, University

More information

ELA THE OPTIMAL PERTURBATION BOUNDS FOR THE WEIGHTED MOORE-PENROSE INVERSE. 1. Introduction. Let C m n be the set of complex m n matrices and C m n

ELA THE OPTIMAL PERTURBATION BOUNDS FOR THE WEIGHTED MOORE-PENROSE INVERSE. 1. Introduction. Let C m n be the set of complex m n matrices and C m n Electronic Journal of Linear Algebra ISSN 08-380 Volume 22, pp. 52-538, May 20 THE OPTIMAL PERTURBATION BOUNDS FOR THE WEIGHTED MOORE-PENROSE INVERSE WEI-WEI XU, LI-XIA CAI, AND WEN LI Abstract. In this

More information

Computation of eigenvalues and singular values Recall that your solutions to these questions will not be collected or evaluated.

Computation of eigenvalues and singular values Recall that your solutions to these questions will not be collected or evaluated. Math 504, Homework 5 Computation of eigenvalues and singular values Recall that your solutions to these questions will not be collected or evaluated 1 Find the eigenvalues and the associated eigenspaces

More information

1 Multiply Eq. E i by λ 0: (λe i ) (E i ) 2 Multiply Eq. E j by λ and add to Eq. E i : (E i + λe j ) (E i )

1 Multiply Eq. E i by λ 0: (λe i ) (E i ) 2 Multiply Eq. E j by λ and add to Eq. E i : (E i + λe j ) (E i ) Direct Methods for Linear Systems Chapter Direct Methods for Solving Linear Systems Per-Olof Persson persson@berkeleyedu Department of Mathematics University of California, Berkeley Math 18A Numerical

More information

STAT 309: MATHEMATICAL COMPUTATIONS I FALL 2017 LECTURE 5

STAT 309: MATHEMATICAL COMPUTATIONS I FALL 2017 LECTURE 5 STAT 39: MATHEMATICAL COMPUTATIONS I FALL 17 LECTURE 5 1 existence of svd Theorem 1 (Existence of SVD) Every matrix has a singular value decomposition (condensed version) Proof Let A C m n and for simplicity

More information

Computing Eigenvalues and/or Eigenvectors;Part 2, The Power method and QR-algorithm

Computing Eigenvalues and/or Eigenvectors;Part 2, The Power method and QR-algorithm Computing Eigenvalues and/or Eigenvectors;Part 2, The Power method and QR-algorithm Tom Lyche Centre of Mathematics for Applications, Department of Informatics, University of Oslo November 13, 2009 Today

More information

On Euclidean distance matrices

On Euclidean distance matrices On Euclidean distance matrices R. Balaji and R. B. Bapat Indian Statistical Institute, New Delhi, 110016 November 19, 2006 Abstract If A is a real symmetric matrix and P is an orthogonal projection onto

More information

arxiv: v1 [math.na] 1 Sep 2018

arxiv: v1 [math.na] 1 Sep 2018 On the perturbation of an L -orthogonal projection Xuefeng Xu arxiv:18090000v1 [mathna] 1 Sep 018 September 5 018 Abstract The L -orthogonal projection is an important mathematical tool in scientific computing

More information

The spectral decomposition of near-toeplitz tridiagonal matrices

The spectral decomposition of near-toeplitz tridiagonal matrices Issue 4, Volume 7, 2013 115 The spectral decomposition of near-toeplitz tridiagonal matrices Nuo Shen, Zhaolin Jiang and Juan Li Abstract Some properties of near-toeplitz tridiagonal matrices with specific

More information

1 Number Systems and Errors 1

1 Number Systems and Errors 1 Contents 1 Number Systems and Errors 1 1.1 Introduction................................ 1 1.2 Number Representation and Base of Numbers............. 1 1.2.1 Normalized Floating-point Representation...........

More information

AMS 209, Fall 2015 Final Project Type A Numerical Linear Algebra: Gaussian Elimination with Pivoting for Solving Linear Systems

AMS 209, Fall 2015 Final Project Type A Numerical Linear Algebra: Gaussian Elimination with Pivoting for Solving Linear Systems AMS 209, Fall 205 Final Project Type A Numerical Linear Algebra: Gaussian Elimination with Pivoting for Solving Linear Systems. Overview We are interested in solving a well-defined linear system given

More information

INTEGER POWERS OF ANTI-BIDIAGONAL HANKEL MATRICES

INTEGER POWERS OF ANTI-BIDIAGONAL HANKEL MATRICES Indian J Pure Appl Math, 49: 87-98, March 08 c Indian National Science Academy DOI: 0007/s36-08-056-9 INTEGER POWERS OF ANTI-BIDIAGONAL HANKEL MATRICES João Lita da Silva Department of Mathematics and

More information

Generalized Principal Pivot Transform

Generalized Principal Pivot Transform Generalized Principal Pivot Transform M. Rajesh Kannan and R. B. Bapat Indian Statistical Institute New Delhi, 110016, India Abstract The generalized principal pivot transform is a generalization of the

More information

Computing tomographic resolution matrices using Arnoldi s iterative inversion algorithm

Computing tomographic resolution matrices using Arnoldi s iterative inversion algorithm Stanford Exploration Project, Report 82, May 11, 2001, pages 1 176 Computing tomographic resolution matrices using Arnoldi s iterative inversion algorithm James G. Berryman 1 ABSTRACT Resolution matrices

More information

Solving large scale eigenvalue problems

Solving large scale eigenvalue problems arge scale eigenvalue problems, Lecture 5, March 23, 2016 1/30 Lecture 5, March 23, 2016: The QR algorithm II http://people.inf.ethz.ch/arbenz/ewp/ Peter Arbenz Computer Science Department, ETH Zürich

More information

JACOBI S ITERATION METHOD

JACOBI S ITERATION METHOD ITERATION METHODS These are methods which compute a sequence of progressively accurate iterates to approximate the solution of Ax = b. We need such methods for solving many large linear systems. Sometimes

More information

Algorithmen zur digitalen Bildverarbeitung I

Algorithmen zur digitalen Bildverarbeitung I ALBERT-LUDWIGS-UNIVERSITÄT FREIBURG INSTITUT FÜR INFORMATIK Lehrstuhl für Mustererkennung und Bildverarbeitung Prof. Dr.-Ing. Hans Burkhardt Georges-Köhler-Allee Geb. 05, Zi 01-09 D-79110 Freiburg Tel.

More information

arxiv: v1 [math.ra] 28 Jan 2016

arxiv: v1 [math.ra] 28 Jan 2016 The Moore-Penrose inverse in rings with involution arxiv:1601.07685v1 [math.ra] 28 Jan 2016 Sanzhang Xu and Jianlong Chen Department of Mathematics, Southeast University, Nanjing 210096, China Abstract:

More information

Recurrence Relations

Recurrence Relations Recurrence Relations Recurrence Relations Reading (Epp s textbook) 5.6 5.8 1 Recurrence Relations A recurrence relation for a sequence aa 0, aa 1, aa 2, ({a n }) is a formula that relates each term a k

More information

Cholesky factorisation of linear systems coming from finite difference approximations of singularly perturbed problems

Cholesky factorisation of linear systems coming from finite difference approximations of singularly perturbed problems Cholesky factorisation of linear systems coming from finite difference approximations of singularly perturbed problems Thái Anh Nhan and Niall Madden Abstract We consider the solution of large linear systems

More information

Group inverse for the block matrix with two identical subblocks over skew fields

Group inverse for the block matrix with two identical subblocks over skew fields Electronic Journal of Linear Algebra Volume 21 Volume 21 2010 Article 7 2010 Group inverse for the block matrix with two identical subblocks over skew fields Jiemei Zhao Changjiang Bu Follow this and additional

More information

Subset selection for matrices

Subset selection for matrices Linear Algebra its Applications 422 (2007) 349 359 www.elsevier.com/locate/laa Subset selection for matrices F.R. de Hoog a, R.M.M. Mattheij b, a CSIRO Mathematical Information Sciences, P.O. ox 664, Canberra,

More information

Partial isometries and EP elements in rings with involution

Partial isometries and EP elements in rings with involution Electronic Journal of Linear Algebra Volume 18 Volume 18 (2009) Article 55 2009 Partial isometries and EP elements in rings with involution Dijana Mosic dragan@pmf.ni.ac.yu Dragan S. Djordjevic Follow

More information

On group inverse of singular Toeplitz matrices

On group inverse of singular Toeplitz matrices Linear Algebra and its Applications 399 (2005) 109 123 wwwelseviercom/locate/laa On group inverse of singular Toeplitz matrices Yimin Wei a,, Huaian Diao b,1 a Department of Mathematics, Fudan Universit,

More information

On the Schur Complement of Diagonally Dominant Matrices

On the Schur Complement of Diagonally Dominant Matrices On the Schur Complement of Diagonally Dominant Matrices T.-G. Lei, C.-W. Woo,J.-Z.Liu, and F. Zhang 1 Introduction In 1979, Carlson and Markham proved that the Schur complements of strictly diagonally

More information

ACM106a - Homework 4 Solutions

ACM106a - Homework 4 Solutions ACM106a - Homework 4 Solutions prepared by Svitlana Vyetrenko November 17, 2006 1. Problem 1: (a) Let A be a normal triangular matrix. Without the loss of generality assume that A is upper triangular,

More information

linearly indepedent eigenvectors as the multiplicity of the root, but in general there may be no more than one. For further discussion, assume matrice

linearly indepedent eigenvectors as the multiplicity of the root, but in general there may be no more than one. For further discussion, assume matrice 3. Eigenvalues and Eigenvectors, Spectral Representation 3.. Eigenvalues and Eigenvectors A vector ' is eigenvector of a matrix K, if K' is parallel to ' and ' 6, i.e., K' k' k is the eigenvalue. If is

More information

Begin accumulation of transformation matrices. This block skipped when i=1. Use u and u/h stored in a to form P Q.

Begin accumulation of transformation matrices. This block skipped when i=1. Use u and u/h stored in a to form P Q. 11.3 Eigenvalues and Eigenvectors of a Tridiagonal Matrix 475 f += e[j]*a[i][j]; hh=f/(h+h); Form K, equation (11.2.11). for (j=1;j

More information

MATH 425-Spring 2010 HOMEWORK ASSIGNMENTS

MATH 425-Spring 2010 HOMEWORK ASSIGNMENTS MATH 425-Spring 2010 HOMEWORK ASSIGNMENTS Instructor: Shmuel Friedland Department of Mathematics, Statistics and Computer Science email: friedlan@uic.edu Last update April 18, 2010 1 HOMEWORK ASSIGNMENT

More information

Improved Newton s method with exact line searches to solve quadratic matrix equation

Improved Newton s method with exact line searches to solve quadratic matrix equation Journal of Computational and Applied Mathematics 222 (2008) 645 654 wwwelseviercom/locate/cam Improved Newton s method with exact line searches to solve quadratic matrix equation Jian-hui Long, Xi-yan

More information

A fast randomized algorithm for orthogonal projection

A fast randomized algorithm for orthogonal projection A fast randomized algorithm for orthogonal projection Vladimir Rokhlin and Mark Tygert arxiv:0912.1135v2 [cs.na] 10 Dec 2009 December 10, 2009 Abstract We describe an algorithm that, given any full-rank

More information

Remarks on the Thickness of K n,n,n

Remarks on the Thickness of K n,n,n Remarks on the Thickness of K n,n,n Yan Yang Department of Mathematics Tianjin University, Tianjin 30007, P.R.China yanyang@tju.edu.cn Abstract The thickness θ(g) of a graph G is the minimum number of

More information

Large Scale Data Analysis Using Deep Learning

Large Scale Data Analysis Using Deep Learning Large Scale Data Analysis Using Deep Learning Linear Algebra U Kang Seoul National University U Kang 1 In This Lecture Overview of linear algebra (but, not a comprehensive survey) Focused on the subset

More information

Operators with Compatible Ranges

Operators with Compatible Ranges Filomat : (7), 579 585 https://doiorg/98/fil7579d Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://wwwpmfniacrs/filomat Operators with Compatible Ranges

More information

Generalized interval arithmetic on compact matrix Lie groups

Generalized interval arithmetic on compact matrix Lie groups myjournal manuscript No. (will be inserted by the editor) Generalized interval arithmetic on compact matrix Lie groups Hermann Schichl, Mihály Csaba Markót, Arnold Neumaier Faculty of Mathematics, University

More information

ELA THE MINIMUM-NORM LEAST-SQUARES SOLUTION OF A LINEAR SYSTEM AND SYMMETRIC RANK-ONE UPDATES

ELA THE MINIMUM-NORM LEAST-SQUARES SOLUTION OF A LINEAR SYSTEM AND SYMMETRIC RANK-ONE UPDATES Volume 22, pp. 480-489, May 20 THE MINIMUM-NORM LEAST-SQUARES SOLUTION OF A LINEAR SYSTEM AND SYMMETRIC RANK-ONE UPDATES XUZHOU CHEN AND JUN JI Abstract. In this paper, we study the Moore-Penrose inverse

More information

Section 5.6. LU and LDU Factorizations

Section 5.6. LU and LDU Factorizations 5.6. LU and LDU Factorizations Section 5.6. LU and LDU Factorizations Note. We largely follow Fraleigh and Beauregard s approach to this topic from Linear Algebra, 3rd Edition, Addison-Wesley (995). See

More information

Interlacing Inequalities for Totally Nonnegative Matrices

Interlacing Inequalities for Totally Nonnegative Matrices Interlacing Inequalities for Totally Nonnegative Matrices Chi-Kwong Li and Roy Mathias October 26, 2004 Dedicated to Professor T. Ando on the occasion of his 70th birthday. Abstract Suppose λ 1 λ n 0 are

More information

Large-scale eigenvalue problems

Large-scale eigenvalue problems ELE 538B: Mathematics of High-Dimensional Data Large-scale eigenvalue problems Yuxin Chen Princeton University, Fall 208 Outline Power method Lanczos algorithm Eigenvalue problems 4-2 Eigendecomposition

More information

Balancing sequences of matrices with application to algebra of balancing numbers

Balancing sequences of matrices with application to algebra of balancing numbers Notes on Number Theory and Discrete Mathematics ISSN 1310 5132 Vol 20 2014 No 1 49 58 Balancing sequences of matrices with application to algebra of balancing numbers Prasanta Kumar Ray International Institute

More information

11.0 Introduction. An N N matrix A is said to have an eigenvector x and corresponding eigenvalue λ if. A x = λx (11.0.1)

11.0 Introduction. An N N matrix A is said to have an eigenvector x and corresponding eigenvalue λ if. A x = λx (11.0.1) Chapter 11. 11.0 Introduction Eigensystems An N N matrix A is said to have an eigenvector x and corresponding eigenvalue λ if A x = λx (11.0.1) Obviously any multiple of an eigenvector x will also be an

More information