On Pth Roots of Stochastic Matrices Higham, Nicholas J
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Tilburg University on the Matrix
Tilburg University On the Matrix (I + X)-1 Engwerda, J.C. Publication date: 2005 Link to publication in Tilburg University Research Portal Citation for published version (APA): Engwerda, J. C. (2005). On the Matrix (I + X)-1. (CentER Discussion Paper; Vol. 2005-120). Macroeconomics. General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal Take down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Download date: 25. sep. 2021 No. 2005–120 -1 ON THE MATRIX INEQUALITY ( I + x ) I . By Jacob Engwerda November 2005 ISSN 0924-7815 On the matrix inequality (I + X)−1 ≤ I. Jacob Engwerda Tilburg University Dept. of Econometrics and O.R. P.O. Box: 90153, 5000 LE Tilburg, The Netherlands e-mail: [email protected] November, 2005 1 Abstract: In this note we consider the question under which conditions all entries of the matrix I −(I +X)−1 are nonnegative in case matrix X is a real positive definite matrix. -
QUADRATIC FORMS and DEFINITE MATRICES 1.1. Definition of A
QUADRATIC FORMS AND DEFINITE MATRICES 1. DEFINITION AND CLASSIFICATION OF QUADRATIC FORMS 1.1. Definition of a quadratic form. Let A denote an n x n symmetric matrix with real entries and let x denote an n x 1 column vector. Then Q = x’Ax is said to be a quadratic form. Note that a11 ··· a1n . x1 Q = x´Ax =(x1...xn) . xn an1 ··· ann P a1ixi . =(x1,x2, ··· ,xn) . P anixi 2 (1) = a11x1 + a12x1x2 + ... + a1nx1xn 2 + a21x2x1 + a22x2 + ... + a2nx2xn + ... + ... + ... 2 + an1xnx1 + an2xnx2 + ... + annxn = Pi ≤ j aij xi xj For example, consider the matrix 12 A = 21 and the vector x. Q is given by 0 12x1 Q = x Ax =[x1 x2] 21 x2 x1 =[x1 +2x2 2 x1 + x2 ] x2 2 2 = x1 +2x1 x2 +2x1 x2 + x2 2 2 = x1 +4x1 x2 + x2 1.2. Classification of the quadratic form Q = x0Ax: A quadratic form is said to be: a: negative definite: Q<0 when x =06 b: negative semidefinite: Q ≤ 0 for all x and Q =0for some x =06 c: positive definite: Q>0 when x =06 d: positive semidefinite: Q ≥ 0 for all x and Q = 0 for some x =06 e: indefinite: Q>0 for some x and Q<0 for some other x Date: September 14, 2004. 1 2 QUADRATIC FORMS AND DEFINITE MATRICES Consider as an example the 3x3 diagonal matrix D below and a general 3 element vector x. 100 D = 020 004 The general quadratic form is given by 100 x1 0 Q = x Ax =[x1 x2 x3] 020 x2 004 x3 x1 =[x 2 x 4 x ] x2 1 2 3 x3 2 2 2 = x1 +2x2 +4x3 Note that for any real vector x =06 , that Q will be positive, because the square of any number is positive, the coefficients of the squared terms are positive and the sum of positive numbers is always positive. -
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. -
Arxiv:1909.13402V1 [Math.CA] 30 Sep 2019 Routh-Hurwitz Array [14], Argument Principle [23] and So On
ON GENERALIZATION OF CLASSICAL HURWITZ STABILITY CRITERIA FOR MATRIX POLYNOMIALS XUZHOU ZHAN AND ALEXANDER DYACHENKO Abstract. In this paper, we associate a class of Hurwitz matrix polynomi- als with Stieltjes positive definite matrix sequences. This connection leads to an extension of two classical criteria of Hurwitz stability for real polynomials to matrix polynomials: tests for Hurwitz stability via positive definiteness of block-Hankel matrices built from matricial Markov parameters and via matricial Stieltjes continued fractions. We obtain further conditions for Hurwitz stability in terms of block-Hankel minors and quasiminors, which may be viewed as a weak version of the total positivity criterion. Keywords: Hurwitz stability, matrix polynomials, total positivity, Markov parameters, Hankel matrices, Stieltjes positive definite sequences, quasiminors 1. Introduction Consider a high-order differential system (n) (n−1) A0y (t) + A1y (t) + ··· + Any(t) = u(t); where A0;:::;An are complex matrices, y(t) is the output vector and u(t) denotes the control input vector. The asymptotic stability of such a system is determined by the Hurwitz stability of its characteristic matrix polynomial n n−1 F (z) = A0z + A1z + ··· + An; or to say, by that all roots of det F (z) lie in the open left half-plane <z < 0. Many algebraic techniques are developed for testing the Hurwitz stability of matrix polynomials, which allow to avoid computing the determinant and zeros: LMI approach [20, 21, 27, 28], the Anderson-Jury Bezoutian [29, 30], matrix Cauchy indices [6], lossless positive real property [4], block Hurwitz matrix [25], extended arXiv:1909.13402v1 [math.CA] 30 Sep 2019 Routh-Hurwitz array [14], argument principle [23] and so on. -
Inner Products and Norms (Part III)
Inner Products and Norms (part III) Prof. Dan A. Simovici UMB 1 / 74 Outline 1 Approximating Subspaces 2 Gram Matrices 3 The Gram-Schmidt Orthogonalization Algorithm 4 QR Decomposition of Matrices 5 Gram-Schmidt Algorithm in R 2 / 74 Approximating Subspaces Definition A subspace T of a inner product linear space is an approximating subspace if for every x 2 L there is a unique element in T that is closest to x. Theorem Let T be a subspace in the inner product space L. If x 2 L and t 2 T , then x − t 2 T ? if and only if t is the unique element of T closest to x. 3 / 74 Approximating Subspaces Proof Suppose that x − t 2 T ?. Then, for any u 2 T we have k x − u k2=k (x − t) + (t − u) k2=k x − t k2 + k t − u k2; by observing that x − t 2 T ? and t − u 2 T and applying Pythagora's 2 2 Theorem to x − t and t − u. Therefore, we have k x − u k >k x − t k , so t is the unique element of T closest to x. 4 / 74 Approximating Subspaces Proof (cont'd) Conversely, suppose that t is the unique element of T closest to x and x − t 62 T ?, that is, there exists u 2 T such that (x − t; u) 6= 0. This implies, of course, that u 6= 0L. We have k x − (t + au) k2=k x − t − au k2=k x − t k2 −2(x − t; au) + jaj2 k u k2 : 2 2 Since k x − (t + au) k >k x − t k (by the definition of t), we have 2 2 1 −2(x − t; au) + jaj k u k > 0 for every a 2 F. -
A Note on Nonnegative Diagonally Dominant Matrices Geir Dahl
UNIVERSITY OF OSLO Department of Informatics A note on nonnegative diagonally dominant matrices Geir Dahl Report 269, ISBN 82-7368-211-0 April 1999 A note on nonnegative diagonally dominant matrices ∗ Geir Dahl April 1999 ∗ e make some observations concerning the set C of real nonnegative, W n diagonally dominant matrices of order . This set is a symmetric and n convex cone and we determine its extreme rays. From this we derive ∗ dierent results, e.g., that the rank and the kernel of each matrix A ∈Cn is , and may b e found explicitly. y a certain supp ort graph of determined b A ∗ ver, the set of doubly sto chastic matrices in C is studied. Moreo n Keywords: Diagonal ly dominant matrices, convex cones, graphs and ma- trices. 1 An observation e recall that a real matrix of order is called diagonal ly dominant if W P A n | |≥ | | for . If all these inequalities are strict, is ai,i j=6 i ai,j i =1,...,n A strictly diagonal ly dominant. These matrices arise in many applications as e.g., discretization of partial dierential equations [14] and cubic spline interp ola- [10], and a typical problem is to solve a linear system where tion Ax = b strictly diagonally dominant, see also [13]. Strict diagonal dominance A is is a criterion which is easy to check for nonsingularity, and this is imp ortant for the estimation of eigenvalues confer Ger²chgorin disks, see e.g. [7]. For more ab out diagonally dominant matrices, see [7] or [13]. A matrix is called nonnegative positive if all its elements are nonnegative p ositive. -
Conditioning Analysis of Positive Definite Matrices by Approximate Factorizations
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Computational and Applied Mathematics 26 (1989) 257-269 257 North-Holland Conditioning analysis of positive definite matrices by approximate factorizations Robert BEAUWENS Service de M&rologie NucGaire, UniversitP Libre de Bruxelles, 50 au. F.D. Roosevelt, 1050 Brussels, Belgium Renaud WILMET The Johns Hopkins University, Baltimore, MD 21218, U.S.A Received 19 February 1988 Revised 7 December 1988 Abstract: The conditioning analysis of positive definite matrices by approximate LU factorizations is usually reduced to that of Stieltjes matrices (or even to more specific classes of matrices) by means of perturbation arguments like spectral equivalence. We show in the present work that a wider class, which we call “almost Stieltjes” matrices, can be used as reference class and that it has decisive advantages for the conditioning analysis of finite element approxima- tions of large multidimensional steady-state diffusion problems. Keywords: Iterative analysis, preconditioning, approximate factorizations, solution of finite element approximations to partial differential equations. 1. Introduction The conditioning analysis of positive definite matrices by approximate factorizations has been the subject of a geometrical approach by Axelsson and his coworkers [1,2,7,8] and of an algebraic approach by one of the authors [3-51. In both approaches, the main role is played by Stieltjes matrices to which other cases are reduced by means of spectral equivalence or by a similar perturbation argument. In applications to discrete partial differential equations, such arguments lead to asymptotic estimates which may be quite poor in nonasymptotic regions, a typical instance in large engineering multidimensional steady-state diffusion applications, as illustrated below. -
Amath/Math 516 First Homework Set Solutions
AMATH/MATH 516 FIRST HOMEWORK SET SOLUTIONS The purpose of this problem set is to have you brush up and further develop your multi-variable calculus and linear algebra skills. The problem set will be very difficult for some and straightforward for others. If you are having any difficulty, please feel free to discuss the problems with me at any time. Don't delay in starting work on these problems! 1. Let Q be an n × n symmetric positive definite matrix. The following fact for symmetric matrices can be used to answer the questions in this problem. Fact: If M is a real symmetric n×n matrix, then there is a real orthogonal n×n matrix U (U T U = I) T and a real diagonal matrix Λ = diag(λ1; λ2; : : : ; λn) such that M = UΛU . (a) Show that the eigenvalues of Q2 are the square of the eigenvalues of Q. Note that λ is an eigenvalue of Q if and only if there is some vector v such that Qv = λv. Then Q2v = Qλv = λ2v, so λ2 is an eigenvalue of Q2. (b) If λ1 ≥ λ2 ≥ · · · ≥ λn are the eigen values of Q, show that 2 T 2 n λnkuk2 ≤ u Qu ≤ λ1kuk2 8 u 2 IR : T T Pn 2 We have u Qu = u UΛUu = i=1 λikuk2. The result follows immediately from bounds on the λi. (c) If 0 < λ < λ¯ are such that 2 T ¯ 2 n λkuk2 ≤ u Qu ≤ λkuk2 8 u 2 IR ; then all of the eigenvalues of Q must lie in the interval [λ; λ¯]. -
Similarities on Graphs: Kernels Versus Proximity Measures1
Similarities on Graphs: Kernels versus Proximity Measures1 Konstantin Avrachenkova, Pavel Chebotarevb,∗, Dmytro Rubanova aInria Sophia Antipolis, France bTrapeznikov Institute of Control Sciences of the Russian Academy of Sciences, Russia Abstract We analytically study proximity and distance properties of various kernels and similarity measures on graphs. This helps to understand the mathemat- ical nature of such measures and can potentially be useful for recommending the adoption of specific similarity measures in data analysis. 1. Introduction Until the 1960s, mathematicians studied only one distance for graph ver- tices, the shortest path distance [6]. In 1967, Gerald Sharpe proposed the electric distance [42]; then it was rediscovered several times. For some col- lection of graph distances, we refer to [19, Chapter 15]. Distances2 are treated as dissimilarity measures. In contrast, similarity measures are maximized when every distance is equal to zero, i.e., when two arguments of the function coincide. At the same time, there is a close relation between distances and certain classes of similarity measures. One of such classes consists of functions defined with the help of kernels on graphs, i.e., positive semidefinite matrices with indices corresponding to the nodes. Every kernel is the Gram matrix of some set of vectors in a Euclidean space, and Schoenberg’s theorem [39, 40] shows how it can be transformed into the matrix of Euclidean distances between these vectors. arXiv:1802.06284v1 [math.CO] 17 Feb 2018 Another class consists of proximity measures characterized by the trian- gle inequality for proximities. Every proximity measure κ(x, y) generates a ∗Corresponding author Email addresses: [email protected] (Konstantin Avrachenkov), [email protected] (Pavel Chebotarev), [email protected] (Dmytro Rubanov) 1This is the authors’ version of a work that was submitted to Elsevier. -
Facts from Linear Algebra
Appendix A Facts from Linear Algebra Abstract We introduce the notation of vector and matrices (cf. Section A.1), and recall the solvability of linear systems (cf. Section A.2). Section A.3 introduces the spectrum σ(A), matrix polynomials P (A) and their spectra, the spectral radius ρ(A), and its properties. Block structures are introduced in Section A.4. Subjects of Section A.5 are orthogonal and orthonormal vectors, orthogonalisation, the QR method, and orthogonal projections. Section A.6 is devoted to the Schur normal form (§A.6.1) and the Jordan normal form (§A.6.2). Diagonalisability is discussed in §A.6.3. Finally, in §A.6.4, the singular value decomposition is explained. A.1 Notation for Vectors and Matrices We recall that the field K denotes either R or C. Given a finite index set I, the linear I space of all vectors x =(xi)i∈I with xi ∈ K is denoted by K . The corresponding square matrices form the space KI×I . KI×J with another index set J describes rectangular matrices mapping KJ into KI . The linear subspace of a vector space V spanned by the vectors {xα ∈V : α ∈ I} is denoted and defined by α α span{x : α ∈ I} := aαx : aα ∈ K . α∈I I×I T Let A =(aαβ)α,β∈I ∈ K . Then A =(aβα)α,β∈I denotes the transposed H matrix, while A =(aβα)α,β∈I is the adjoint (or Hermitian transposed) matrix. T H Note that A = A holds if K = R . Since (x1,x2,...) indicates a row vector, T (x1,x2,...) is used for a column vector. -
Similar Matrices and Jordan Form
Similar matrices and Jordan form We’ve nearly covered the entire heart of linear algebra – once we’ve finished singular value decompositions we’ll have seen all the most central topics. AT A is positive definite A matrix is positive definite if xT Ax > 0 for all x 6= 0. This is a very important class of matrices; positive definite matrices appear in the form of AT A when computing least squares solutions. In many situations, a rectangular matrix is multiplied by its transpose to get a square matrix. Given a symmetric positive definite matrix A, is its inverse also symmet ric and positive definite? Yes, because if the (positive) eigenvalues of A are −1 l1, l2, · · · ld then the eigenvalues 1/l1, 1/l2, · · · 1/ld of A are also positive. If A and B are positive definite, is A + B positive definite? We don’t know much about the eigenvalues of A + B, but we can use the property xT Ax > 0 and xT Bx > 0 to show that xT(A + B)x > 0 for x 6= 0 and so A + B is also positive definite. Now suppose A is a rectangular (m by n) matrix. A is almost certainly not symmetric, but AT A is square and symmetric. Is AT A positive definite? We’d rather not try to find the eigenvalues or the pivots of this matrix, so we ask when xT AT Ax is positive. Simplifying xT AT Ax is just a matter of moving parentheses: xT (AT A)x = (Ax)T (Ax) = jAxj2 ≥ 0. -
Inverse M-Matrix, a New Characterization
Linear Algebra and its Applications 595 (2020) 182–191 Contents lists available at ScienceDirect Linear Algebra and its Applications www.elsevier.com/locate/laa Inverse M-matrix, a new characterization Claude Dellacherie a, Servet Martínez b, Jaime San Martín b,∗ a Laboratoire Raphaël Salem, UMR 6085, Université de Rouen, Site du Madrillet, 76801 Saint Étienne du Rouvray Cedex, France b CMM-DIM, Universidad de Chile, UMI-CNRS 2807, Casilla 170-3 Correo 3, Santiago, Chile a r t i c l e i n f o a b s t r a c t Article history: In this article we present a new characterization of inverse M- Received 18 June 2019 matrices, inverse row diagonally dominant M-matrices and Accepted 20 February 2020 inverse row and column diagonally dominant M-matrices, Available online 22 February 2020 based on the positivity of certain inner products. Submitted by P. Semrl © 2020 Elsevier Inc. All rights reserved. MSC: 15B35 15B51 60J10 Keywords: M-matrix Inverse M-matrix Potentials Complete Maximum Principle Markov chains Contents 1. Introduction and main results.......................................... 183 2. Proof of Theorem 1.1 and Theorem 1.3 .................................... 186 3. Some complements.................................................. 189 * Corresponding author. E-mail addresses: [email protected] (C. Dellacherie), [email protected] (S. Martínez), [email protected] (J. San Martín). https://doi.org/10.1016/j.laa.2020.02.024 0024-3795/© 2020 Elsevier Inc. All rights reserved. C. Dellacherie et al. / Linear Algebra and its Applications 595 (2020) 182–191 183 Acknowledgement....................................................... 190 References............................................................ 190 1. Introduction and main results In this short note, we give a new characterization of inverses M-matrices, inverses of row diagonally dominant M-matrices and inverses of row and column diagonally domi- nant M-matrices.