The QR Decomposition
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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. -
QR Decomposition: History and Its Applications
Mathematics & Statistics Auburn University, Alabama, USA QR History Dec 17, 2010 Asymptotic result QR iteration QR decomposition: History and its EE Applications Home Page Title Page Tin-Yau Tam èèèUUUÎÎÎ JJ II J I Page 1 of 37 Æâ§w Go Back fÆêÆÆÆ Full Screen Close email: [email protected] Website: www.auburn.edu/∼tamtiny Quit 1. QR decomposition Recall the QR decomposition of A ∈ GLn(C): QR A = QR History Asymptotic result QR iteration where Q ∈ GLn(C) is unitary and R ∈ GLn(C) is upper ∆ with positive EE diagonal entries. Such decomposition is unique. Set Home Page a(A) := diag (r11, . , rnn) Title Page where A is written in column form JJ II J I A = (a1| · · · |an) Page 2 of 37 Go Back Geometric interpretation of a(A): Full Screen rii is the distance (w.r.t. 2-norm) between ai and span {a1, . , ai−1}, Close i = 2, . , n. Quit Example: 12 −51 4 6/7 −69/175 −58/175 14 21 −14 6 167 −68 = 3/7 158/175 6/175 0 175 −70 . QR −4 24 −41 −2/7 6/35 −33/35 0 0 35 History Asymptotic result QR iteration EE • QR decomposition is the matrix version of the Gram-Schmidt orthonor- Home Page malization process. Title Page JJ II • QR decomposition can be extended to rectangular matrices, i.e., if A ∈ J I m×n with m ≥ n (tall matrix) and full rank, then C Page 3 of 37 A = QR Go Back Full Screen where Q ∈ Cm×n has orthonormal columns and R ∈ Cn×n is upper ∆ Close with positive “diagonal” entries. -
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 . -
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). -
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. -
Lecture 4: Applications of Orthogonality: QR Decompositions Week 4 UCSB 2014
Math 108B Professor: Padraic Bartlett Lecture 4: Applications of Orthogonality: QR Decompositions Week 4 UCSB 2014 In our last class, we described the following method for creating orthonormal bases, known as the Gram-Schmidt method: ~ ~ Theorem. Suppose that V is a k-dimensional space with a basis B = fb1;::: bkg. The following process (called the Gram-Schmidt process) creates an orthonormal basis for V : 1. First, create the following vectors f ~v1; : : : ~vkg: • ~u1 = b~1: • ~u2 = b~2 − proj(b~2 onto ~u1). • ~u3 = b~3 − proj(b~3 onto ~u1) − proj(b~3 onto ~u2). • ~u4 = b~4 − proj(b~4 onto ~u1) − proj(b~4 onto ~u2) − proj(b~4 onto ~u3). ~ ~ ~ • ~uk = bk − proj(bk onto ~u1) − ::: − proj(bk onto k~u −1). 2. Now, take each of the vectors ~ui, and rescale them so that they are unit length: i.e. ~ui redefine each ~ui as the rescaled vector . jj ~uijj In this class, I want to talk about a useful application of the Gram-Schmidt method: the QR decomposition! We define this here: 1 The QR decomposition Definition. We say that an n×n matrix Q is orthogonal if its columns form an orthonor- n mal basis for R . As a side note: we've studied these matrices before! In class on Friday, we proved that for any such matrix, the relation QT · Q = I held; to see this, we just looked at the (i; j)-th entry of the product QT · Q, which by definition was the i-th row of QT dotted with the j-th column of A. -
Numerical Linear Algebra Revised February 15, 2010 4.1 the LU
Numerical Linear Algebra Revised February 15, 2010 4.1 The LU Decomposition The Elementary Matrices and badgauss In the previous chapters we talked a bit about the solving systems of the form Lx = b and Ux = b where L is lower triangular and U is upper triangular. In the exercises for Chapter 2 you were asked to write a program x=lusolver(L,U,b) which solves LUx = b using forsub and backsub. We now address the problem of representing a matrix A as a product of a lower triangular L and an upper triangular U: Recall our old friend badgauss. function B=badgauss(A) m=size(A,1); B=A; for i=1:m-1 for j=i+1:m a=-B(j,i)/B(i,i); B(j,:)=a*B(i,:)+B(j,:); end end The heart of badgauss is the elementary row operation of type 3: B(j,:)=a*B(i,:)+B(j,:); where a=-B(j,i)/B(i,i); Note also that the index j is greater than i since the loop is for j=i+1:m As we know from linear algebra, an elementary opreration of type 3 can be viewed as matrix multipli- cation EA where E is an elementary matrix of type 3. E looks just like the identity matrix, except that E(j; i) = a where j; i and a are as in the MATLAB code above. In particular, E is a lower triangular matrix, moreover the entries on the diagonal are 1's. We call such a matrix unit lower triangular. -
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. -
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. -
A Fast and Accurate Matrix Completion Method Based on QR Decomposition and L 2,1-Norm Minimization Qing Liu, Franck Davoine, Jian Yang, Ying Cui, Jin Zhong, Fei Han
A Fast and Accurate Matrix Completion Method based on QR Decomposition and L 2,1-Norm Minimization Qing Liu, Franck Davoine, Jian Yang, Ying Cui, Jin Zhong, Fei Han To cite this version: Qing Liu, Franck Davoine, Jian Yang, Ying Cui, Jin Zhong, et al.. A Fast and Accu- rate Matrix Completion Method based on QR Decomposition and L 2,1-Norm Minimization. IEEE Transactions on Neural Networks and Learning Systems, IEEE, 2019, 30 (3), pp.803-817. 10.1109/TNNLS.2018.2851957. hal-01927616 HAL Id: hal-01927616 https://hal.archives-ouvertes.fr/hal-01927616 Submitted on 20 Nov 2018 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. Page 1 of 20 1 1 2 3 A Fast and Accurate Matrix Completion Method 4 5 based on QR Decomposition and L -Norm 6 2,1 7 8 Minimization 9 Qing Liu, Franck Davoine, Jian Yang, Member, IEEE, Ying Cui, Zhong Jin, and Fei Han 10 11 12 13 14 Abstract—Low-rank matrix completion aims to recover ma- I. INTRODUCTION 15 trices with missing entries and has attracted considerable at- tention from machine learning researchers. -
Notes on Orthogonal Projections, Operators, QR Factorizations, Etc. These Notes Are Largely Based on Parts of Trefethen and Bau’S Numerical Linear Algebra
Notes on orthogonal projections, operators, QR factorizations, etc. These notes are largely based on parts of Trefethen and Bau’s Numerical Linear Algebra. Please notify me of any typos as soon as possible. 1. Basic lemmas and definitions Throughout these notes, V is a n-dimensional vector space over C with a fixed Her- mitian product. Unless otherwise stated, it is assumed that bases are chosen so that the Hermitian product is given by the conjugate transpose, i.e. that (x, y) = y∗x where y∗ is the conjugate transpose of y. Recall that a projection is a linear operator P : V → V such that P 2 = P . Its complementary projection is I − P . In class we proved the elementary result that: Lemma 1. Range(P ) = Ker(I − P ), and Range(I − P ) = Ker(P ). Another fact worth noting is that Range(P ) ∩ Ker(P ) = {0} so the vector space V = Range(P ) ⊕ Ker(P ). Definition 1. A projection is orthogonal if Range(P ) is orthogonal to Ker(P ). Since it is convenient to use orthonormal bases, but adapt them to different situations, we often need to change basis in an orthogonality-preserving way. I.e. we need the following type of operator: Definition 2. A nonsingular matrix Q is called an unitary matrix if (x, y) = (Qx, Qy) for every x, y ∈ V . If Q is real, it is called an orthogonal matrix. An orthogonal matrix Q can be thought of as an orthogonal change of basis matrix, in which each column is a new basis vector. Lemma 2. -
GRAM SCHMIDT and QR FACTORIZATION Math 21B, O. Knill
GRAM SCHMIDT AND QR FACTORIZATION Math 21b, O. Knill GRAM-SCHMIDT PROCESS. Let ~v1; : : : ;~vn be a basis in V . Let ~w1 = ~v1 and ~u1 = ~w1=jj~w1jj. The Gram-Schmidt process recursively constructs from the already constructed orthonormal set ~u1; : : : ; ~ui−1 which spans a linear space Vi−1 the new vector ~wi = (~vi − projVi−1 (~vi)) which is orthogonal to Vi−1, and then normalizes ~wi to get ~ui = ~wi=jj~wijj. Each vector ~wi is orthonormal to the linear space Vi−1. The vectors f~u1; : : : ; ~un g form then an orthonormal basis in V . EXAMPLE. 2 2 3 2 1 3 2 1 3 Find an orthonormal basis for ~v1 = 4 0 5, ~v2 = 4 3 5 and ~v3 = 4 2 5. 0 0 5 SOLUTION. 2 1 3 1. ~u1 = ~v1=jj~v1jj = 4 0 5. 0 2 0 3 2 0 3 3 1 2. ~w2 = (~v2 − projV1 (~v2)) = ~v2 − (~u1 · ~v2)~u1 = 4 5. ~u2 = ~w2=jj~w2jj = 4 5. 0 0 2 0 3 2 0 3 0 0 3. ~w3 = (~v3 − projV2 (~v3)) = ~v3 − (~u1 · ~v3)~u1 − (~u2 · ~v3)~u2 = 4 5, ~u3 = ~w3=jj~w3jj = 4 5. 5 1 QR FACTORIZATION. The formulas can be written as ~v1 = jj~v1jj~u1 = r11~u1 ··· ~vi = (~u1 · ~vi)~u1 + ··· + (~ui−1 · ~vi)~ui−1 + jj~wijj~ui = r1i~u1 + ··· + rii~ui ··· ~vn = (~u1 · ~vn)~u1 + ··· + (~un−1 · ~vn)~un−1 + jj~wnjj~un = r1n~u1 + ··· + rnn~un which means in matrix form 2 3 2 3 2 3 j j · j j j · j r11 r12 · r1n A = 4 ~v1 ···· ~vn 5 = 4 ~u1 ···· ~un 5 4 0 r22 · r2n 5 = QR ; j j · j j j · j 0 0 · rnn where A and Q are m × n matrices and R is a n × n matrix which has rij = ~vj · ~ui, for i < j and vii = j~vij THE GRAM-SCHMIDT PROCESS PROVES: Any matrix A with linearly independent columns ~vi can be decomposed as A = QR, where Q has orthonormal column vectors and where R is an upper triangular square matrix with the same number of columns than A.