DOCSLIB.ORG

Linear Algebra Review

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Linear Algebra Review Appendix A Linear Algebra Review In this appendix, we review results from linear algebra that are used in the text. The results quoted here are mostly standard, and the proofs are mostly omitted. For more information, the reader is encouraged to consult such standard linear algebra textbooks as [HK]or[Axl]. Throughout this appendix, we let Mn.C/ denote the space of n n matrices with entries in C: A.1 Eigenvectors and Eigenvalues n For any A 2 Mn.C/; a nonzero vector v in C is called an eigenvector for A if there is some complex number such that Av D v: An eigenvalue for A is a complex number for which there exists a nonzero v 2 Cn with Av D v: Thus, is an eigenvalue for A if the equation Av D v or, equivalently, the equation .A I /v D 0; has a nonzero solution v: This happens precisely when A I fails to be invertible, which is precisely when det.A I / D 0: For any A 2 Mn.C/; the characteristic polynomial p of A is given by A previous version of this book was inadvertently published without the middle initial of the author’s name as “Brian Hall”. For this reason an erratum has been published, correcting the mistake in the previous version and showing the correct name as Brian C. Hall (see DOI http:// dx.doi.org/10.1007/978-3-319-13467-3_14). The version readers currently see is the corrected version. The Publisher would like to apologize for the earlier mistake. © Springer International Publishing Switzerland 2015 409 B.C. Hall, Lie Groups, Lie Algebras, and Representations, Graduate Texts in Mathematics 222, DOI 10.1007/978-3-319-13467-3 410 A Linear Algebra Review p./ D det.A I /; 2 C: This polynomial has degree n: In light of the preceding discussion, the eigenvalues are precisely the zeros of the characteristic polynomial. We can define, more generally, the notion of eigenvector and eigenvalue for any linear operator on a vector space. If V is a finite-dimensional vector space over C (or over any algebraically closed field), every linear operator on V will have a least one eigenvalue. If A is a linear operator on a vector space V and is an eigenvalue for A; the -eigenspace for A; denoted V; is the set of all vectors v 2 V (including the zero vector) that satisfy Av D v: The -eigenspace for A is a subspace of V: The dimension of this space is called the multiplicity of : (More precisely, this is the “geometric multiplicity” of : In the finite-dimensional case, there is also a notion of the “algebraic multiplicity” of ; which is the number of times that occurs as a root of the characteristic polynomial. The geometric multiplicity of cannot exceed the algebraic multiplicity). Proposition A.1. Suppose that A is a linear operator on a vector space V and v1;:::;vk are eigenvectors with distinct eigenvalues 1;:::;k: Then v1;:::;vk are linearly independent. Note that here V does not have to be finite dimensional. Proposition A.2. Suppose that A and B are linear operators on a finite- dimensional vector space V and suppose that AB D BA: Then for each eigenvalue of A; the operator B maps the -eigenspace of A into itself. Proof. Let be an eigenvalue of A and let V be the -eigenspace of A: Then let v be an element of V and consider Bv: Since B commutes with A; we have A.Bv/ D BAv D Bv, showing that Bv is in V: ut A.2 Diagonalization Two matrices A; B 2 Mn.C/ are said to be similar if there exists an invertible matrix C such that A D CBC 1: The operation B ! CBC 1 is called conjugation of B by C: A matrix is said to be diagonalizable if it is similar to a diagonal matrix. A matrix A 2 Mn.C/ is diagonalizable if and only if there exist n linearly independent eigenvectors for A: Specifically, if v1;:::;vn are linearly independent eigenvectors, let C be the matrix whose kth column is vk: Then C is invertible and we will have A.3 Generalized Eigenvectors and the SN Decomposition 411 0 1 1 B : C 1 A D C @ :: A C ; (A.1) n where 1;:::;n are the eigenvalues associated to the eigenvectors v1;:::;vn; in that order. To verify (A.1), note that C maps the standard basis element ej to vj : 1 Thus, C maps vj to ej ; the diagonal matrix on the right-hand side of (A.1)then maps ej to j ej ; and C maps j ej to j vj : Thus, both sides of (A.1)mapvj to j vj ; for all j: If A 2 Mn.C/ has n distinct eigenvalues (i.e., n distinct roots to the characteristic polynomial), A is necessarily diagonalizable, by Proposition A.1. If the characteris- tic polynomial of A has repeated roots, A may or may not be diagonalizable. For A 2 Mn.C/; the adjoint of A; denoted A ; is the conjugate-transpose of A; .A /jk D Akj: (A.2) AmatrixA is said to be self-adjoint (or Hermitian)ifA D A: AmatrixA is said to be skew self-adjoint (or skew Hermitian)ifA DA: A matrix is said to be unitary if A D A1: More generally, A is said to be normal if A commutes with A: If A is normal, A is necessarily diagonalizable, and, indeed, it is possible to find an orthonormal basis of eigenvectors for A: In such cases, the matrix C in (A.1) may be taken to be unitary. If A is self-adjoint, all of its eigenvalues are real. If A is real and self-adjoint (or, equivalently, real and symmetric), the eigenvectors may be taken to be real as well, which means that in this case, the matrix C may be taken to be orthogonal. If A is skew, then its eigenvalues are pure imaginary. If A is unitary, then its eigenvalues are complex numbers of absolute value 1. We summarize the results of the previous paragraphs in the following. Theorem A.3. Suppose that A 2 Mn.C/ has the property that A A D AA ; (e.g., if A D A; A D A1; or A DA). Then A is diagonalizable and it is possible to find an orthonormal basis for Cn consisting of eigenvectors for A: If A D A; all the eigenvalues of A are real; if A DA; all the eigenvalues of A are pure imaginary; and if A D A1; all the eigenvalues of A have absolute value 1. A.3 Generalized Eigenvectors and the SN Decomposition Not all matrices are diagonalizable, even over C: If, for example,  à 11 A D ; (A.3) 01 412 A Linear Algebra Review then the only eigenvalue of A is 1, and every eigenvector with eigenvalue 1 is of the form .c; 0/: Thus, we cannot find two linearly independent eigenvectors for A: It is not hard, however, to prove the following result. Recall that a matrix A is nilpotent if Ak D 0 for some positive integer k: Theorem A.4. Every matrix is similar to an upper triangular matrix. Every nilpotent matrix is similar to an upper triangular matrix with zeros on the diagonal. While Theorem A.4 is sufficient for some purposes, we will in general need something that comes a bit closer to a diagonal representation. If A 2 Mn.C/ does not have n linearly independent eigenvectors, we may consider the more general concept of generalized eigenvectors. A nonzero vector v 2 Cn is called a generalized eigenvector for A if there is some complex number and some positive integer k such that .A I /kv D 0: (A.4) If (A.4) holds for some v ¤ 0,then.AI / cannot be invertible. Thus, the number must be an (ordinary) eigenvalue for A: However, for a fixed eigenvalue ; there may be generalized eigenvectors v that are not ordinary eigenvectors. In the case of the matrix A in (A.3), for example, the vector .0; 1/ is a generalized eigenvector with eigenvalue 1 (with k D 2). It can be shown that every A 2 Mn.C/ has a basis of generalized eigenvectors. For any matrix A and any eigenvalue for A; let W be the generalized eigenspace with eigenvalue : ˇ n ˇ k W Dfv 2 C .A I / v D 0 for some k g: n Then C decomposes as a direct sum of the W’s, as ranges over all the eigenvalues of A: Furthermore, the subspace W is easily seen to be invariant under the matrix A.LetA denote the restriction of A to the subspace W; and let N D A I; so that A D I C N: k Then N is nilpotent; that is, N D 0 for some positive integer k: We summarize the preceding discussion in the following theorem. Theorem A.5. Let A be an n n complex matrix. Then there exists a basis for Cn consisting of generalized eigenvectors for A: Furthermore, Cn is the direct sum of the generalized eigenspaces W: Each W is invariant under A; and the restriction of A to W is of the form I C N; where N is nilpotent. The preceding result is the basis for the following decomposition. Theorem A.6. Each A 2 Mn.C/ has a unique decomposition as A D S CN where S is diagonalizable, N is nilpotent, and SN D NS: A.5 The Trace 413 The expression A D S C N; with S and N as in the theorem, is called the SN decomposition of A: The existence of an SN decomposition follows from the previous theorem: We define S to be the operator equal to I on each generalized eigenspace W of A and we set N to be the operator equal to N on each W.For example, if A is the matrix in (A.3), then we have  à  à 10 01 S D ;ND : 01 00 A.4 The Jordan Canonical Form The Jordan canonical form may be viewed as a refinement of the SN decomposition, based on a further analysis of the nilpotent matrices N in Theorem A.5.
Load more

Recommended publications
  • Lecture Notes in One File
    Phys 6124 zipped! World Wide Quest to Tame Math Methods Predrag Cvitanovic´ January 2, 2020 Overview The career of a young theoretical physicist consists of treating the harmonic oscillator in ever-increasing levels of abstraction. — Sidney Coleman I am leaving the course notes here, not so much for the notes themselves –they cannot be understood on their own, without the (unrecorded) live lectures– but for the hyperlinks to various source texts you might find useful later on in your research. We change the topics covered year to year, in hope that they reflect better what a graduate student needs to know. This year’s experiment are the two weeks dedicated to data analysis. Let me know whether you would have preferred the more traditional math methods fare, like Bessels and such. If you have taken this course live, you might have noticed a pattern: Every week we start with something obvious that you already know, let mathematics lead us on, and then suddenly end up someplace surprising and highly non-intuitive. And indeed, in the last lecture (that never took place), we turn Coleman on his head, and abandon harmonic potential for the inverted harmonic potential, “spatiotemporal cats” and chaos, arXiv:1912.02940. Contents 1 Linear algebra7 HomeworkHW1 ................................ 7 1.1 Literature ................................ 8 1.2 Matrix-valued functions ......................... 9 1.3 A linear diversion ............................ 12 1.4 Eigenvalues and eigenvectors ...................... 14 1.4.1 Yes, but how do you really do it? . 18 References ................................... 19 exercises 20 2 Eigenvalue problems 23 HomeworkHW2 ................................ 23 2.1 Normal modes .............................. 24 2.2 Stable/unstable manifolds .......................
    [Show full text]
  • Introduction to Quantum Field Theory
    Utrecht Lecture Notes Masters Program Theoretical Physics September 2006 INTRODUCTION TO QUANTUM FIELD THEORY by B. de Wit Institute for Theoretical Physics Utrecht University Contents 1 Introduction 4 2 Path integrals and quantum mechanics 5 3 The classical limit 10 4 Continuous systems 18 5 Field theory 23 6 Correlation functions 36 6.1 Harmonic oscillator correlation functions; operators . 37 6.2 Harmonic oscillator correlation functions; path integrals . 39 6.2.1 Evaluating G0 ............................... 41 6.2.2 The integral over qn ............................ 42 6.2.3 The integrals over q1 and q2 ....................... 43 6.3 Conclusion . 44 7 Euclidean Theory 46 8 Tunneling and instantons 55 8.1 The double-well potential . 57 8.2 The periodic potential . 66 9 Perturbation theory 71 10 More on Feynman diagrams 80 11 Fermionic harmonic oscillator states 87 12 Anticommuting c-numbers 90 13 Phase space with commuting and anticommuting coordinates and quanti- zation 96 14 Path integrals for fermions 108 15 Feynman diagrams for fermions 114 2 16 Regularization and renormalization 117 17 Further reading 127 3 1 Introduction Physical systems that involve an infinite number of degrees of freedom are usually described by some sort of field theory. Almost all systems in nature involve an extremely large number of degrees of freedom. A droplet of water contains of the order of 1026 molecules and while each water molecule can in many applications be described as a point particle, each molecule has itself a complicated structure which reveals itself at molecular length scales. To deal with this large number of degrees of freedom, which for all practical purposes is infinite, we often regard a system as continuous, in spite of the fact that, at certain distance scales, it is discrete.
    [Show full text]
  • Journal of Computational and Applied Mathematics Exponentials of Skew
    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 233 (2010) 2867–2875 Contents lists available at ScienceDirect Journal of Computational and Applied Mathematics journal homepage: www.elsevier.com/locate/cam Exponentials of skew-symmetric matrices and logarithms of orthogonal matrices João R. Cardoso a,b,∗, F. Silva Leite c,b a Institute of Engineering of Coimbra, Rua Pedro Nunes, 3030-199 Coimbra, Portugal b Institute of Systems and Robotics, University of Coimbra, Pólo II, 3030-290 Coimbra, Portugal c Department of Mathematics, University of Coimbra, 3001-454 Coimbra, Portugal article info a b s t r a c t Article history: Two widely used methods for computing matrix exponentials and matrix logarithms are, Received 11 July 2008 respectively, the scaling and squaring and the inverse scaling and squaring. Both methods become effective when combined with Padé approximation. This paper deals with the MSC: computation of exponentials of skew-symmetric matrices and logarithms of orthogonal 65Fxx matrices. Our main goal is to improve these two methods by exploiting the special structure Keywords: of skew-symmetric and orthogonal matrices. Geometric features of the matrix exponential Skew-symmetric matrix and logarithm and extensions to the special Euclidean group of rigid motions are also Orthogonal matrix addressed. Matrix exponential ' 2009 Elsevier B.V. All rights reserved. Matrix logarithm Scaling and squaring Inverse scaling and squaring Padé approximants 1. Introduction n×n Given a square matrix X 2 R , the exponential of X is given by the absolute convergent power series 1 X X k eX D : kW kD0 n×n X Reciprocally, given a nonsingular matrix A 2 R , any solution of the matrix equation e D A is called a logarithm of A.
    [Show full text]
  • The Exponential Map for the Group of Similarity Transformations and Applications to Motion Interpolation
    The exponential map for the group of similarity transformations and applications to motion interpolation Spyridon Leonardos1, Christine Allen–Blanchette2 and Jean Gallier3 Abstract— In this paper, we explore the exponential map The problem of motion interpolation has been studied and its inverse, the logarithm map, for the group SIM(n) of extensively both in the robotics and computer graphics n similarity transformations in R which are the composition communities. Since rotations can be represented by quater- of a rotation, a translation and a uniform scaling. We give a formula for the exponential map and we prove that it is nions, the problem of quaternion interpolation has been surjective. We give an explicit formula for the case of n = 3 investigated, an approach initiated by Shoemake [21], [22], and show how to efficiently compute the logarithmic map. As who extended the de Casteljau algorithm to the 3-sphere. an application, we use these algorithms to perform motion Related work was done by Barr et al. [2]. Moreover, Kim interpolation. Given a sequence of similarity transformations, et al. [13], [14] corrected bugs in Shoemake and introduced we compute a sequence of logarithms, then fit a cubic spline 4 that interpolates the logarithms and finally, we compute the various kinds of splines on the 3-sphere in R , using the interpolating curve in SIM(3). exponential map. Motion interpolation and rational motions have been investigated by Juttler¨ [9], [10], Juttler¨ and Wagner I. INTRODUCTION [11], [12], Horsch and Juttler¨ [8], and Roschel¨ [20]. Park and Ravani [18], [19] also investigated Bezier´ curves on The exponential map is ubiquitous in engineering appli- Riemannian manifolds and Lie groups and in particular, the cations ranging from dynamical systems and robotics to group of rotations.
    [Show full text]
  • Arxiv:2004.07192V2 [Quant-Ph] 14 May 2021
    Two-way covert quantum communication in the microwave regime R. Di Candia,1, ∗ H. Yi˘gitler,1 G. S. Paraoanu,2 and R. J¨antti1 1Department of Communications and Networking, Aalto University, Espoo, 02150 Finland 2QTF Centre of Excellence, Department of Applied Physics, Aalto University School of Science, FI-00076 AALTO, Finland Quantum communication addresses the problem of exchanging information across macroscopic distances by employing encryption techniques based on quantum mechanical laws. Here, we ad- vance a new paradigm for secure quantum communication by combining backscattering concepts with covert communication in the microwave regime. Our protocol allows communication between Alice, who uses only discrete phase modulations, and Bob, who has access to cryogenic microwave technology. Using notions of quantum channel discrimination and quantum metrology, we find the ultimate bounds for the receiver performance, proving that quantum correlations can enhance the signal-to-noise ratio by up to 6 dB. These bounds rule out any quantum illumination advantage when the source is strongly amplified, and shows that a relevant gain is possible only in the low photon-number regime. We show how the protocol can be used for covert communication, where the carrier signal is indistinguishable from the thermal noise in the environment. We complement our information-theoretic results with a feasible experimental proposal in a circuit QED platform. This work makes a decisive step toward implementing secure quantum communication concepts in the previously uncharted 1 − 10 GHz frequency range, in the scenario when the disposable power of one party is severely constrained. I. INTRODUCTION range. For instance, QKD security proofs require level of noises which at room temperature are reachable only by frequencies at least in the Terahertz band [8].
    [Show full text]
  • Randomized Linear Algebra Approaches to Estimate the Von Neumann Entropy of Density Matrices
    Randomized Linear Algebra Approaches to Estimate the Von Neumann Entropy of Density Matrices Eugenia-Maria Kontopoulou Ananth Grama Wojciech Szpankowski Petros Drineas Purdue University Purdue University Purdue University Purdue University Computer Science Computer Science Computer Science Computer Science West Lafayette, IN, USA West Lafayette, IN, USA West Lafayette, IN, USA West Lafayette, IN, USA Email: [email protected] Email: [email protected] Email: [email protected] Email: [email protected] Abstract—The von Neumann entropy, named after John von Motivated by the above discussion, we seek numerical Neumann, is the extension of classical entropy concepts to the algorithms that approximate the von Neumann entropy of large field of quantum mechanics and, from a numerical perspective, density matrices, e.g., symmetric positive definite matrices can be computed simply by computing all the eigenvalues of a 3 density matrix, an operation that could be prohibitively expensive with unit trace, faster than the trivial O(n ) approach. Our for large-scale density matrices. We present and analyze two algorithms build upon recent developments in the field of randomized algorithms to approximate the von Neumann entropy Randomized Numerical Linear Algebra (RandNLA), an in- of density matrices: our algorithms leverage recent developments terdisciplinary research area that exploits randomization as a in the Randomized Numerical Linear Algebra (RandNLA) liter- computational resource to develop improved algorithms for ature, such as randomized trace
    [Show full text]
  • Computing Real Square Roots of a Real Matrix* LINEAR ALGEBRA
    Computing Real Square Roots of a Real Matrix* Nicholas J. Higham Department of Mathematics University of Manchester Manchester Ml3 9PL, England In memory of James H. WiIkinson Submitted by Hans Schneider ABSTRACT Bjiirck and Hammarling [l] describe a fast, stable Schur method for computing a square root X of a matrix A (X2 = A).We present an extension of their method which enables real arithmetic to be used throughout when computing a real square root of a real matrix. For a nonsingular real matrix A conditions are given for the existence of a real square root, and for the existence of a real square root which is a polynomial in A; thenumber of square roots of the latter type is determined. The conditioning of matrix square roots is investigated, and an algorithm is given for the computation of a well-conditioned square root. 1. INTRODUCTION Given a matrix A, a matrix X for which X2 = A is called a square root of A. Several authors have considered the computation of matrix square roots [3, 4, 9, 10, 15, 161. A particularly attractive method which utilizes the Schur decomposition is described by Bjiirck and Hammarling [l]; in general it requires complex arithmetic. Our main purpose is to show how the method can be extended so as to compute a real square root of a real matrix, if one exists, in real arithmetic. The theory behind the existence of matrix square roots is nontrivial, as can be seen by noting that while the n x n identity matrix has infinitely many square roots for n > 2 (any involutary matrix such as a Householder transformation is a square root), a nonsingular Jordan block has precisely two square roots (this is proved in Corollary 1).
    [Show full text]
  • Floquet Theory for Time Scales and Putzer Representations of Matrix Logarithms
    Taylor & Francis Journal of Difference Equations and Applications, Ta@rhFrancir tmup 2003 VOL.9 (1). pp. 77-92 Floquet Theory for Time Scales and Putzer Representations of Matrix Logarithms CALVIN D. AHLBRANDP,* and JERRY RIDENHOURb" aDepartment of Mathematics, University of Missouri, Columbia, MO 6521 1, USA; bDepartment of Mathematics and Statistics, Utah State University, Logan, UT 84322-3900, USA (Received 14 September 2001; Revised 9 October 2001; In final form 9 October 2001) Dedicated to Allan Peterson on the occasion of his 60th birthday. A Floquet theory is presented that generalizes known results for differential equations and for difference equations to the setting of dynamic equations on time scales. Since logarithms of matrices play a key role in Floquet theory, considerable effort is expended in producing case-free exact representations of the principal branch of the matrix logarithm. Such representations were fmt produced by Putzer as representations of matrix exponentials. Some representations depend on knowledge of the eigenvalues while others depend only on the coefficients of the characteristic polynomial. Logarithms of special forms of matrices are also considered. In particular, it is shown that the square of a nonsingular matrix with real entries has a logarithm with real entries. Keywords: Floquet theory; Matrix logarithm; Principal branch; Putzer representation 2000 AMS Subject ClassiJication: 39A10 1. INTRODUCTION The theory of dynamic equations on time scales (or, more generally, measure chains) was introduced in Stefan Hilger's PhD thesis in 1988. The theory presents a structure where, once a result is established for general time scales, special cases include a result for differential equations (obtained by taking the time scale to be the real numbers) and a result for difference equations (obtained by taking the time scale to be the integers).
    [Show full text]
  • Fast Computation of Von Neumann Entropy for Large-Scale Graphs Via Quadratic Approximations
    Linear Algebra and its Applications 585 (2020) 127–146 Contents lists available at ScienceDirect Linear Algebra and its Applications www.elsevier.com/locate/laa Fast computation of von Neumann entropy for large-scale graphs via quadratic approximations Hayoung Choi a,∗, Jinglian He b, Hang Hu b, Yuanming Shi b a Research Institute of Mathematics, Seoul National University, Republic of Korea b School of Information Science and Technology, ShanghaiTech University, China a r t i c l e i n f o a b s t r a c t Article history: The von Neumann graph entropy (VNGE) can be used as Received 21 November 2018 a measure of graph complexity, which can be the measure Accepted 24 September 2019 of information divergence and distance between graphs. Available online 1 October 2019 However, computing VNGE is extensively demanding for a Submitted by M. Benzi large-scale graph. We propose novel quadratic approximations MSC: for fast computing VNGE. Various inequalities for error 05C12 between the quadratic approximations and the exact VNGE 05C50 are found. Our methods reduce the cubic complexity of VNGE 05C80 to linear complexity. Computational simulations on random 05C90 graph models and various real network datasets demonstrate 81P45 superior performance. 81P68 © 2019 Elsevier Inc. All rights reserved. Keywords: Von Neumann entropy Von Neumann graph entropy Graph dissimilarity Density matrix 1. Introduction Graph is one of the most common representations of complex data. It has the sophis- ticated capability of representing and summarizing irregular structural features. Today, * Corresponding author. E-mail addresses: [email protected] (H. Choi), [email protected] (J.
    [Show full text]
  • Logarithms and Square Roots of Real Matrices Existence, Uniqueness, and Applications in Medical Imaging
    Logarithms and Square Roots of Real Matrices Existence, Uniqueness, and Applications in Medical Imaging Jean Gallier Department of Computer and Information Science University of Pennsylvania Philadelphia, PA 19104, USA [email protected] September 2, 2019 Abstract. The need for computing logarithms or square roots of real matrices arises in a number of applied problems. A significant class of problems comes from medical imag- ing. One of these problems is to interpolate and to perform statistics on data represented by certain kinds of matrices (such as symmetric positive definite matrices in DTI). Another important and difficult problem is the registration of medical images. For both of these prob- lems, the ability to compute logarithms of real matrices turns out to be crucial. However, not all real matrices have a real logarithm and thus, it is important to have sufficient conditions for the existence (and possibly the uniqueness) of a real logarithm for a real matrix. Such conditions (involving the eigenvalues of a matrix) are known, both for the logarithm and the square root. As far as I know, with the exception of Higham's recent book [18], proofs of the results involving these conditions are scattered in the literature and it is not easy to locate them. Moreover, Higham's excellent book assumes a certain level of background in linear algebra that readers interested in applications to medical imaging may not possess so we feel that a more elementary presentation might be a valuable supplement to Higham [18]. In this paper, I present a unified exposition of these results, including a proof of the existence of the Real Jordan Form, and give more direct proofs of some of these results using the Real Jordan Form.
    [Show full text]
  • Bayesian Inference of Log Determinants
    Bayesian Inference of Log Determinants Jack Fitzsimons1 Kurt Cutajar2 Michael Osborne1 Stephen Roberts1 Maurizio Filippone2 1 Information Engineering, University of Oxford, UK 2 Department of Data Science, EURECOM, France Abstract The standard approach for evaluating the log- determinant of a positive definite matrix involves The log-determinant of a kernel matrix ap- the use of Cholesky decomposition (Golub & Van Loan, pears in a variety of machine learning prob- 1996), which is employed in various applications of lems, ranging from determinantal point pro- statistical models such as kernel machines. However, cesses and generalized Markov random fields, the use of Cholesky decomposition for general dense 3 through to the training of Gaussian processes. matrices requires O(n ) operations, whilst also entail- 2 Exact calculation of this term is often in- ing memory requirements of O(n ). In view of this tractable when the size of the kernel matrix ex- computational bottleneck, various models requiring ceeds a few thousands. In the spirit of proba- the log-determinant for inference bypass the need to bilistic numerics, we reinterpret the problem of compute it altogether (Anitescu et al., 2012; Stein et al., computing the log-determinant as a Bayesian 2013; Cutajar et al., 2016; Filippone & Engler, 2015). inference problem. In particular, we com- Alternatively, several methods exploit sparsity and struc- bine prior knowledge in the form of bounds ture within the matrix itself to accelerate computations. from matrix theory and evidence derived from For example, sparsity in Gaussian Markov Random stochastic trace estimation to obtain proba- fields (GMRFs) arises from encoding conditional inde- bilistic estimates for the log-determinant and pendence assumptions that are readily available when its associated uncertainty within a given com- considering low-dimensional problems.
    [Show full text]
  • Means and Averaging in the Group of Rotations∗
    To appear in: SIAM J. MATRIX ANAL. APPL. MEANS AND AVERAGING IN THE GROUP OF ROTATIONS∗ MAHER MOAKHERy Abstract. In this paper we give precise definitions of different, properly invariant notions of mean or average rotation. Each mean is associated with a metric in SO(3). The metric induced from the Frobenius inner product gives rise to a mean rotation that is given by the closest special orthogonal matrix to the usual arithmetic mean of the given rotation matrices. The mean rotation associated with the intrinsic metric on SO(3) is the Riemannian center of mass of the given rotation matrices. We show that the Riemannian mean rotation shares many common features with the geometric mean of positive numbers and the geometric mean of positive Hermitian operators. We give some examples with closed-form solutions of both notions of mean. Key words. Special orthogonal group, Rotation, Geodesics, Operator means, Averaging AMS subject classifications. 47A64, 65F30 1. Introduction. In many applications, such as the study of plate tectonics [22] or sequence-dependent continuum modeling of DNA [19], experimental data are given as a sequence of three-dimensional orientation data that usually contain a substantial amount of noise. A common problem is to remove or reduce the noise by processing the raw data, for example by the construction of a suitable filter, in order to obtain appropriately smooth data. Three-dimensional orientation data are elements of the group of rotations that generally are given as a sequence of proper orthogonal matrices, or a sequence of Euler angles, or a sequence of unit quaternions, etc.
    [Show full text]