Kronecker Product of Networked Systems and their Approximates
|
|
- Heather Ball
- 5 years ago
- Views:
Transcription
1 Kronecker Product of Networked Systems and their Approximates Robotics, Aerospace and Information Networks (RAIN) University of Washington (Robotics, Aerospace Kronecker and Information Products Networks (RAIN)) University of Washington 1 / 19
2 Graph Products: Networks within Networks Many ways to compose graphs G and H Cartesian product G H Tensor/Direct/Kronecker product G H Strong product G H Lexicographic product G H Rooted product G H Corona product G H Star product G H How does modularity of the network manifest itself as modularity within the state dynamics? Kronecker Product: (Graphs, ) (Dynamics, ) (Robotics, Aerospace Kronecker and Information Products Networks (RAIN)) University of Washington 2 / 19
3 Graph Product Examples Periodic Structures: e.g., hypercube multiprocessors, building trusses Compartmental Networks: e.g., air traffic networks, chemical reactions Constant degree expander graphs: e.g., computer networks, sorting networks, cryptography Australian Academic Research Network (AARNET) Cite: Parsonage et al. Generalized Graph Products for Network Design and Analysis, (Robotics, Aerospace Kronecker and Information Products Networks (RAIN)) University of Washington 3 / 19
4 Graph Kronecker Product Kronecker product G H Vertex set: V (G H) = V (G) V (H) Edge set: (x 1,x 2 ) (y 1,y 2 ) is in G H if x 1 y 1 and x 2 y 2 Algebraic Representation A(G H) = A(G) A(H) (Robotics, Aerospace Kronecker and Information Products Networks (RAIN)) University of Washington 4 / 19
5 Graph Kronecker Product Weighted: Directed: Multiple Products: (Robotics, Aerospace Kronecker and Information Products Networks (RAIN)) University of Washington 5 / 19
6 Relevance and Intuition: Fractal Nature Recursive growth of graph communities: Nodes get expanded to micro communities A(K) A(K K) A(K K K) A(K K K K) Obey common network features: Degree distribution, density power law, diameters, spectra [Leskovec et al. 10] (Robotics, Aerospace Kronecker and Information Products Networks (RAIN)) University of Washington 6 / 19
7 Relevance and Intuition: Null-Models Attribute representations, e.g., High/Low GPA, Year in school Nodes describe attributes, e.g., u = (High GPA,Yr 12), v = (Low GPA,Yr 11) Edges describe interaction probabilities, e.g., p(u v) = = 0.02 GPA High Low High Low Class Yr12 Yr11 Yr Yr A(G 1 ) A(G 2 ) (GPA,Class) (High,Yr12) (High,Yr11) (Low,Yr12) (Low,Yr11) (High,Yr12) (High,Yr11) (Low,Yr12) (Low,Yr11) A(G 1 G 2 ) (Robotics, Aerospace Kronecker and Information Products Networks (RAIN)) University of Washington 7 / 19
8 Dynamics over Kronecker Product The factor dynamics for i = 1,2,... Consider the discrete dynamics x i (k + 1) = A(G i )x i (k) y i (k) = C i x i (k) x(k + 1) = A(G 1 G 2...)x(k) = A( G i )x(k) y(k) = (C 1 C 2...)x(k) = C i x(k) For output node sets S 1,S 2,... then C(S 1 ) C(S 2 ) = C(S 1 S 2...) Here A( ) preserved the Kronecker product, e.g., Adjacency A(G 1 ), Row stochastic adjacency [A s (G)] ij = [A(G)] ij j [A(G)] ij How does the features of the factors compare to the composite system? (Robotics, Aerospace Kronecker and Information Products Networks (RAIN)) University of Washington 8 / 19
9 Trajectory and Stability When initialized from x(0) = x i (0) the composite trajectory is x(k + 1) = x i (k) Consequence: If the factors are stable then the composite is stable (Robotics, Aerospace Kronecker and Information Products Networks (RAIN)) University of Washington 9 / 19
10 Observability Dynamics are observable if for any unknown x(0), t f there exists a t f such that knowledges of u(t) and y(t) over [0,t 1 ] uniquely determine x(0). Significant in networked robotic systems, human-swarm interaction, network security, quantum networks. Challenging to establish for large networks Known families of observable graphs for selected outputs Paths (Rahmani and Mesbahi 07) Circulants (Nabi-Abdolyousefi and Mesbahi 12) Grids (Parlengeli and Notarsefano 11) Distance regular graphs (Zhang et al. 11) Cartesian products (Chapman et al. 14) (Robotics, Aerospace Kronecker and Information Products Networks (RAIN)) University of Washington 10 / 19
11 Observability Consider diagonalizable A(G 1 ) and A(G 2 ) with distinct eigenvalues of λ 1,..., λ p and µ 1,..., µ q Theorem The pair (A(G 1 G 2 ),C 1 C 2 ) is observable if and only if (1) the pairs (A(G 1 ),C 1 ) and (A(G 2 ),C 2 ) are observable and (2) for λ 1 µ 1 = λ 2 µ 2 = = λ p µ p, λ i λ j i j, p > 1, C T 1 [U 1,U 2,...,U p ] and/or C T 2 [V 1,V 2,...,V p ], where the columns of U i are the orthogonal right eigenvectors of eigenvalues λ i of A(G 1 ) (sim. for pairs ( µ i,v i ) of A(G 2 )). If the factors are observable then all modes such that λ i µ s λ j µ t are observable If (1) and (2) satisfied, and C 1 and C 2 are minimal rank observable on the factors then C 1 C 2 is a minimal rank observable on the composite (Robotics, Aerospace Kronecker and Information Products Networks (RAIN)) University of Washington 11 / 19
12 Observability Example (A(G 1 ),C(S 1 )) is observable with S 1 = {blue, green} (A(G 2 ),C(S 2 )) is observable with S 2 = { } No new multiplicities are introduced in A(G 1 G 2 ) = (A(G 1 G 2 ),C(S 1 S 2 )) is observable with S 1 S 2 = {blue, green } (Robotics, Aerospace Kronecker and Information Products Networks (RAIN)) University of Washington 12 / 19
13 Observability Factorization - Idea of the Proof Popov-Belevitch-Hautus (PBH) test (A, C) is unobservable if and only if there exists a right eigenvalue-eigenvector pair (λ,v) of A such that Cv = 0. Eigenvalue and eigenvector relationship: A(G 1 ) A(G 2 ) A(G 1 G 2 ) Eigenvalue λ i µ j λ i µ j Eigenvector v i u j v i u j Also (C 1 C 2 )(v i u i ) = C 1 v i C 2 u i The proof for simple eigenvalues follows from these observations. (Robotics, Aerospace Kronecker and Information Products Networks (RAIN)) University of Washington 13 / 19
14 Graph Factorization A graph can be factored as well as composed... Theorem (Sabidussi 1960) Every connected graph can be factored as a Kronecker product of prime graphs. This is NOT unique up to reordering of the factors. Primes: G = G 1 G 2 implies that either G 1 or G 2 is K 1 Number of prime factors is at most log G Algorithms Van Loan and Pitsianis (1993) - Exact and an approximation O(n 3 ) (more later) Leskovec et al. (2010) - KronFit approximation of the form G 1 G 1 G 1 (Robotics, Aerospace Kronecker and Information Products Networks (RAIN)) University of Washington 14 / 19
15 Kronecker Product Approximations Van Loan and Pitsianis developed an efficient method to solve min A A 1 A 2 2,F, i.e., the closest Kronecker product G G 1 G 2 G 1 G 2 G 1 G 2 G How does the features of these approximate factors compare to the composite system? (Robotics, Aerospace Kronecker and Information Products Networks (RAIN)) University of Washington 15 / 19
16 Trajectory Approximation Approximate Kronecker Dynamics x(k + 1) = (A(G 1 G 2 ) + )x(k) y(k) = (C 1 C 2 )x(k) For unforced dynamics if x(0) = x 1 (0) x 2 (0) then the trajectory can be approximated by x a (k) = x 1 (k) x 2 (k) where A = A(G 1 G 2 ) A x(k) x a (k) x(0) I k A I k A I A I (Robotics, Aerospace Kronecker and Information Products Networks (RAIN)) University of Washington 16 / 19
17 Distance to Instability Distance to instability is d A = inf( : A + is unstable) = 1 ρ(d A ). If the distance to instability of the factors is d G1 and d G2 then the distance to instability of the composite is d G1 G 2 = d G1 + d G2 d G1 d G2 Consequence: A stable composite dynamics is always more stable than its factors dynamics, i.e., d G1 G 2 max(d G1,d G2 ) (Robotics, Aerospace Kronecker and Information Products Networks (RAIN)) University of Washington 17 / 19
18 Distance to Unobservability Distance to unobservability is d A,C = inf { : (A +,C) is unobservable } Let λ 1 and µ 1 are the smallest magnitude eigenvalues of A(G 1 ) and A(G 2 ), respectively If the distance to unobservability of the factors is d G1,C 1 and d G2,C 2 then the distance to unobservability of the composite is bounded as d G1 G 2,C min( λ 1 d G2,C 2, µ 1 d G1,C 1 ) Consequence: For factor dynamics with a stable mode, the composite dynamics is always closer to unobservability than the factors, i.e., d G1 G 2,C min(d G2,C 2,d G1,C 1 ) (Robotics, Aerospace Kronecker and Information Products Networks (RAIN)) University of Washington 18 / 19
19 Conclusion Kronecker product dynamics related to its factors examining Trajectory Stability Observability Approximate Kronecker dynamics related to its approximate factors examining Bounded Trajectory Distance to instability Distance to unobservability Future work: Lower bounds for distance to unobservability Input/Output approximations Powers of Kronecker products (Robotics, Aerospace Kronecker and Information Products Networks (RAIN)) University of Washington 19 / 19
Cartesian Products of Z-Matrix Networks: Factorization and Interval Analysis
Nabi-Abdolyouse) Factorization University (Distributed andofinterval Washington Space Analysis Systems1 Lab / 27(D Cartesian Products of Z-Matrix Networks: Factorization and Interval Analysis Airlie Chapman
More informationState Controllability, Output Controllability and Stabilizability of Networks: A Symmetry Perspective
State Controllability, Output Controllability and Stabilizability of Networks: A Symmetry Perspective Airlie Chapman and Mehran Mesbahi Robotics, Aerospace, and Information Networks Lab (RAIN) University
More informationOn Symmetry and Controllability of Multi-Agent Systems
53rd IEEE Conference on Decision and Control December 15-17, 2014. Los Angeles, California, USA On Symmetry and Controllability of Multi-Agent Systems Airlie Chapman and Mehran Mesbahi Abstract This paper
More information1 Similarity transform 2. 2 Controllability The PBH test for controllability Observability The PBH test for observability...
Contents 1 Similarity transform 2 2 Controllability 3 21 The PBH test for controllability 5 3 Observability 6 31 The PBH test for observability 7 4 Example ([1, pp121) 9 5 Subspace decomposition 11 51
More informationControl Systems. System response. L. Lanari
Control Systems m i l e r p r a in r e v y n is o System response L. Lanari Outline What we are going to see: how to compute in the s-domain the forced response (zero-state response) using the transfer
More informationOn almost equitable partitions and network controllability
2016 American Control Conference (ACC) Boston Marriott Copley Place July 6-8, 2016. Boston, MA, USA On almost equitable partitions and network controllability Cesar O. Aguilar 1 and Bahman Gharesifard
More informationControl Systems. Laplace domain analysis
Control Systems Laplace domain analysis L. Lanari outline introduce the Laplace unilateral transform define its properties show its advantages in turning ODEs to algebraic equations define an Input/Output
More informationSection 5.4 (Systems of Linear Differential Equation); 9.5 Eigenvalues and Eigenvectors, cont d
Section 5.4 (Systems of Linear Differential Equation); 9.5 Eigenvalues and Eigenvectors, cont d July 6, 2009 Today s Session Today s Session A Summary of This Session: Today s Session A Summary of This
More informationAn Algorithmist s Toolkit September 10, Lecture 1
18.409 An Algorithmist s Toolkit September 10, 2009 Lecture 1 Lecturer: Jonathan Kelner Scribe: Jesse Geneson (2009) 1 Overview The class s goals, requirements, and policies were introduced, and topics
More information1 Continuous-time Systems
Observability Completely controllable systems can be restructured by means of state feedback to have many desirable properties. But what if the state is not available for feedback? What if only the output
More informationObservability. Dynamic Systems. Lecture 2 Observability. Observability, continuous time: Observability, discrete time: = h (2) (x, u, u)
Observability Dynamic Systems Lecture 2 Observability Continuous time model: Discrete time model: ẋ(t) = f (x(t), u(t)), y(t) = h(x(t), u(t)) x(t + 1) = f (x(t), u(t)), y(t) = h(x(t)) Reglerteknik, ISY,
More informationMachine Learning for Data Science (CS4786) Lecture 11
Machine Learning for Data Science (CS4786) Lecture 11 Spectral clustering Course Webpage : http://www.cs.cornell.edu/courses/cs4786/2016sp/ ANNOUNCEMENT 1 Assignment P1 the Diagnostic assignment 1 will
More informationHW2 - Due 01/30. Each answer must be mathematically justified. Don t forget your name.
HW2 - Due 0/30 Each answer must be mathematically justified. Don t forget your name. Problem. Use the row reduction algorithm to find the inverse of the matrix 0 0, 2 3 5 if it exists. Double check your
More informationMATH 221, Spring Homework 10 Solutions
MATH 22, Spring 28 - Homework Solutions Due Tuesday, May Section 52 Page 279, Problem 2: 4 λ A λi = and the characteristic polynomial is det(a λi) = ( 4 λ)( λ) ( )(6) = λ 6 λ 2 +λ+2 The solutions to the
More informationMa/CS 6b Class 23: Eigenvalues in Regular Graphs
Ma/CS 6b Class 3: Eigenvalues in Regular Graphs By Adam Sheffer Recall: The Spectrum of a Graph Consider a graph G = V, E and let A be the adjacency matrix of G. The eigenvalues of G are the eigenvalues
More informationMULTIVARIABLE ZEROS OF STATE-SPACE SYSTEMS
Copyright F.L. Lewis All rights reserved Updated: Monday, September 9, 8 MULIVARIABLE ZEROS OF SAE-SPACE SYSEMS If a system has more than one input or output, it is called multi-input/multi-output (MIMO)
More informationLinear Algebra (MATH ) Spring 2011 Final Exam Practice Problem Solutions
Linear Algebra (MATH 4) Spring 2 Final Exam Practice Problem Solutions Instructions: Try the following on your own, then use the book and notes where you need help. Afterwards, check your solutions with
More informationDiffusion and random walks on graphs
Diffusion and random walks on graphs Leonid E. Zhukov School of Data Analysis and Artificial Intelligence Department of Computer Science National Research University Higher School of Economics Structural
More information1 Some Facts on Symmetric Matrices
1 Some Facts on Symmetric Matrices Definition: Matrix A is symmetric if A = A T. Theorem: Any symmetric matrix 1) has only real eigenvalues; 2) is always iagonalizable; 3) has orthogonal eigenvectors.
More informationControl and synchronization in systems coupled via a complex network
Control and synchronization in systems coupled via a complex network Chai Wah Wu May 29, 2009 2009 IBM Corporation Synchronization in nonlinear dynamical systems Synchronization in groups of nonlinear
More information16.31 Fall 2005 Lecture Presentation Mon 31-Oct-05 ver 1.1
16.31 Fall 2005 Lecture Presentation Mon 31-Oct-05 ver 1.1 Charles P. Coleman October 31, 2005 1 / 40 : Controllability Tests Observability Tests LEARNING OUTCOMES: Perform controllability tests Perform
More informationOn the reachability and observability. of path and cycle graphs
On the reachability and observability 1 of path and cycle graphs Gianfranco Parlangeli Giuseppe Notarstefano arxiv:1109.3556v1 [math.oc] 16 Sep 2011 Abstract In this paper we investigate the reachability
More informationNetworks as vectors of their motif frequencies and 2-norm distance as a measure of similarity
Networks as vectors of their motif frequencies and 2-norm distance as a measure of similarity CS322 Project Writeup Semih Salihoglu Stanford University 353 Serra Street Stanford, CA semih@stanford.edu
More informationCS224W: Social and Information Network Analysis Jure Leskovec, Stanford University
CS224W: Social and Information Network Analysis Jure Leskovec Stanford University Jure Leskovec, Stanford University http://cs224w.stanford.edu Task: Find coalitions in signed networks Incentives: European
More informationMath 18, Linear Algebra, Lecture C00, Spring 2017 Review and Practice Problems for Final Exam
Math 8, Linear Algebra, Lecture C, Spring 7 Review and Practice Problems for Final Exam. The augmentedmatrix of a linear system has been transformed by row operations into 5 4 8. Determine if the system
More informationMarkov Chains and Spectral Clustering
Markov Chains and Spectral Clustering Ning Liu 1,2 and William J. Stewart 1,3 1 Department of Computer Science North Carolina State University, Raleigh, NC 27695-8206, USA. 2 nliu@ncsu.edu, 3 billy@ncsu.edu
More informationStructural Controllability and Observability of Linear Systems Over Finite Fields with Applications to Multi-Agent Systems
Structural Controllability and Observability of Linear Systems Over Finite Fields with Applications to Multi-Agent Systems Shreyas Sundaram, Member, IEEE, and Christoforos N. Hadjicostis, Senior Member,
More informationMining of Massive Datasets Jure Leskovec, AnandRajaraman, Jeff Ullman Stanford University
Note to other teachers and users of these slides: We would be delighted if you found this our material useful in giving your own lectures. Feel free to use these slides verbatim, or to modify them to fit
More informationSome constructions of integral graphs
Some constructions of integral graphs A. Mohammadian B. Tayfeh-Rezaie School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran ali m@ipm.ir tayfeh-r@ipm.ir
More informationMa/CS 6b Class 20: Spectral Graph Theory
Ma/CS 6b Class 20: Spectral Graph Theory By Adam Sheffer Eigenvalues and Eigenvectors A an n n matrix of real numbers. The eigenvalues of A are the numbers λ such that Ax = λx for some nonzero vector x
More informationarxiv: v1 [math.oc] 12 Feb 2018
Controllability Analysis of Threshold Graphs and Cographs Shima Sadat Mousavi Mohammad Haeri and Mehran Mesbahi arxiv:80204022v [mathoc] 2 Feb 208 Abstract In this paper we investigate the controllability
More informationSingular Value Decomposition
Singular Value Decomposition Motivatation The diagonalization theorem play a part in many interesting applications. Unfortunately not all matrices can be factored as A = PDP However a factorization A =
More informationNetwork Reconstruction from Intrinsic Noise: Non-Minimum-Phase Systems
Preprints of the 19th World Congress he International Federation of Automatic Control Network Reconstruction from Intrinsic Noise: Non-Minimum-Phase Systems David Hayden, Ye Yuan Jorge Goncalves Department
More informationAn Algorithmist s Toolkit September 15, Lecture 2
18.409 An Algorithmist s Toolkit September 15, 007 Lecture Lecturer: Jonathan Kelner Scribe: Mergen Nachin 009 1 Administrative Details Signup online for scribing. Review of Lecture 1 All of the following
More informationMATH 423 Linear Algebra II Lecture 20: Geometry of linear transformations. Eigenvalues and eigenvectors. Characteristic polynomial.
MATH 423 Linear Algebra II Lecture 20: Geometry of linear transformations. Eigenvalues and eigenvectors. Characteristic polynomial. Geometric properties of determinants 2 2 determinants and plane geometry
More informationarxiv: v1 [math.oc] 15 Feb 2019
Generalizing Laplacian Controllability of Paths arxiv:1902.05671v1 [math.oc] 15 Feb 2019 Abstract Shun-Pin Hsu, Ping-Yen Yang Department of Electrical Engineering, National Chung Hsing University 250,
More informationStructural Consensus Controllability of Singular Multi-agent Linear Dynamic Systems
Structural Consensus Controllability of Singular Multi-agent Linear Dynamic Systems M. ISAL GARCÍA-PLANAS Universitat Politècnica de Catalunya Departament de Matèmatiques Minería 1, sc. C, 1-3, 08038 arcelona
More informationLinear Algebra. Shan-Hung Wu. Department of Computer Science, National Tsing Hua University, Taiwan. Large-Scale ML, Fall 2016
Linear Algebra Shan-Hung Wu shwu@cs.nthu.edu.tw Department of Computer Science, National Tsing Hua University, Taiwan Large-Scale ML, Fall 2016 Shan-Hung Wu (CS, NTHU) Linear Algebra Large-Scale ML, Fall
More informationThe Hypercube Graph and the Inhibitory Hypercube Network
The Hypercube Graph and the Inhibitory Hypercube Network Michael Cook mcook@forstmannleff.com William J. Wolfe Professor of Computer Science California State University Channel Islands william.wolfe@csuci.edu
More informationDesigning Information Devices and Systems II Fall 2015 Note 22
EE 16B Designing Information Devices and Systems II Fall 2015 Note 22 Notes taken by John Noonan (11/12) Graphing of the State Solutions Open loop x(k + 1) = Ax(k) + Bu(k) y(k) = Cx(k) Closed loop x(k
More informationFinal Exam, Linear Algebra, Fall, 2003, W. Stephen Wilson
Final Exam, Linear Algebra, Fall, 2003, W. Stephen Wilson Name: TA Name and section: NO CALCULATORS, SHOW ALL WORK, NO OTHER PAPERS ON DESK. There is very little actual work to be done on this exam if
More informationName Solutions Linear Algebra; Test 3. Throughout the test simplify all answers except where stated otherwise.
Name Solutions Linear Algebra; Test 3 Throughout the test simplify all answers except where stated otherwise. 1) Find the following: (10 points) ( ) Or note that so the rows are linearly independent, so
More informationAPPLICATIONS The eigenvalues are λ = 5, 5. An orthonormal basis of eigenvectors consists of
CHAPTER III APPLICATIONS The eigenvalues are λ =, An orthonormal basis of eigenvectors consists of, The eigenvalues are λ =, A basis of eigenvectors consists of, 4 which are not perpendicular However,
More informationKronecker Product Approximation with Multiple Factor Matrices via the Tensor Product Algorithm
Kronecker Product Approximation with Multiple actor Matrices via the Tensor Product Algorithm King Keung Wu, Yeung Yam, Helen Meng and Mehran Mesbahi Department of Mechanical and Automation Engineering,
More informationDIAGONALIZATION. In order to see the implications of this definition, let us consider the following example Example 1. Consider the matrix
DIAGONALIZATION Definition We say that a matrix A of size n n is diagonalizable if there is a basis of R n consisting of eigenvectors of A ie if there are n linearly independent vectors v v n such that
More information18.312: Algebraic Combinatorics Lionel Levine. Lecture 19
832: Algebraic Combinatorics Lionel Levine Lecture date: April 2, 20 Lecture 9 Notes by: David Witmer Matrix-Tree Theorem Undirected Graphs Let G = (V, E) be a connected, undirected graph with n vertices,
More informationData Mining and Analysis: Fundamental Concepts and Algorithms
: Fundamental Concepts and Algorithms dataminingbook.info Mohammed J. Zaki 1 Wagner Meira Jr. 2 1 Department of Computer Science Rensselaer Polytechnic Institute, Troy, NY, USA 2 Department of Computer
More information22m:033 Notes: 7.1 Diagonalization of Symmetric Matrices
m:33 Notes: 7. Diagonalization of Symmetric Matrices Dennis Roseman University of Iowa Iowa City, IA http://www.math.uiowa.edu/ roseman May 3, Symmetric matrices Definition. A symmetric matrix is a matrix
More informationMa/CS 6b Class 20: Spectral Graph Theory
Ma/CS 6b Class 20: Spectral Graph Theory By Adam Sheffer Recall: Parity of a Permutation S n the set of permutations of 1,2,, n. A permutation σ S n is even if it can be written as a composition of an
More informationRadial eigenvectors of the Laplacian of the nonbinary hypercube
Radial eigenvectors of the Laplacian of the nonbinary hypercube Murali K. Srinivasan Department of Mathematics Indian Institute of Technology, Bombay Powai, Mumbai 400076, INDIA mks@math.iitb.ac.in murali.k.srinivasan@gmail.com
More informationEE226a - Summary of Lecture 13 and 14 Kalman Filter: Convergence
1 EE226a - Summary of Lecture 13 and 14 Kalman Filter: Convergence Jean Walrand I. SUMMARY Here are the key ideas and results of this important topic. Section II reviews Kalman Filter. A system is observable
More informationMATH 20F: LINEAR ALGEBRA LECTURE B00 (T. KEMP)
MATH 20F: LINEAR ALGEBRA LECTURE B00 (T KEMP) Definition 01 If T (x) = Ax is a linear transformation from R n to R m then Nul (T ) = {x R n : T (x) = 0} = Nul (A) Ran (T ) = {Ax R m : x R n } = {b R m
More informationChapter 11. Matrix Algorithms and Graph Partitioning. M. E. J. Newman. June 10, M. E. J. Newman Chapter 11 June 10, / 43
Chapter 11 Matrix Algorithms and Graph Partitioning M. E. J. Newman June 10, 2016 M. E. J. Newman Chapter 11 June 10, 2016 1 / 43 Table of Contents 1 Eigenvalue and Eigenvector Eigenvector Centrality The
More informationControllability, Observability, Full State Feedback, Observer Based Control
Multivariable Control Lecture 4 Controllability, Observability, Full State Feedback, Observer Based Control John T. Wen September 13, 24 Ref: 3.2-3.4 of Text Controllability ẋ = Ax + Bu; x() = x. At time
More informationCoding the Matrix Index - Version 0
0 vector, [definition]; (2.4.1): 68 2D geometry, transformations in, [lab]; (4.15.0): 196-200 A T (matrix A transpose); (4.5.4): 157 absolute value, complex number; (1.4.1): 43 abstract/abstracting, over
More informationResearch Article Mathematical Model and Cluster Synchronization for a Complex Dynamical Network with Two Types of Chaotic Oscillators
Applied Mathematics Volume 212, Article ID 936, 12 pages doi:1.11/212/936 Research Article Mathematical Model and Cluster Synchronization for a Complex Dynamical Network with Two Types of Chaotic Oscillators
More informationSymmetric 0-1 matrices with inverses having two distinct values and constant diagonal
Symmetric 0-1 matrices with inverses having two distinct values and constant diagonal Wayne Barrett Department of Mathematics, Brigham Young University, Provo, UT, 84602, USA Steve Butler Dept. of Mathematics,
More informationControl Systems Design
ELEC4410 Control Systems Design Lecture 14: Controllability Julio H. Braslavsky julio@ee.newcastle.edu.au School of Electrical Engineering and Computer Science Lecture 14: Controllability p.1/23 Outline
More informationLinear Algebra Final Exam Study Guide Solutions Fall 2012
. Let A = Given that v = 7 7 67 5 75 78 Linear Algebra Final Exam Study Guide Solutions Fall 5 explain why it is not possible to diagonalize A. is an eigenvector for A and λ = is an eigenvalue for A diagonalize
More informationMath 304 Handout: Linear algebra, graphs, and networks.
Math 30 Handout: Linear algebra, graphs, and networks. December, 006. GRAPHS AND ADJACENCY MATRICES. Definition. A graph is a collection of vertices connected by edges. A directed graph is a graph all
More informationLab 8: Measuring Graph Centrality - PageRank. Monday, November 5 CompSci 531, Fall 2018
Lab 8: Measuring Graph Centrality - PageRank Monday, November 5 CompSci 531, Fall 2018 Outline Measuring Graph Centrality: Motivation Random Walks, Markov Chains, and Stationarity Distributions Google
More informationDS-GA 1002 Lecture notes 0 Fall Linear Algebra. These notes provide a review of basic concepts in linear algebra.
DS-GA 1002 Lecture notes 0 Fall 2016 Linear Algebra These notes provide a review of basic concepts in linear algebra. 1 Vector spaces You are no doubt familiar with vectors in R 2 or R 3, i.e. [ ] 1.1
More informationModule 03 Linear Systems Theory: Necessary Background
Module 03 Linear Systems Theory: Necessary Background Ahmad F. Taha EE 5243: Introduction to Cyber-Physical Systems Email: ahmad.taha@utsa.edu Webpage: http://engineering.utsa.edu/ taha/index.html September
More informationECE557 Systems Control
ECE557 Systems Control Bruce Francis Course notes, Version.0, September 008 Preface This is the second Engineering Science course on control. It assumes ECE56 as a prerequisite. If you didn t take ECE56,
More informationMath Camp Lecture 4: Linear Algebra. Xiao Yu Wang. Aug 2010 MIT. Xiao Yu Wang (MIT) Math Camp /10 1 / 88
Math Camp 2010 Lecture 4: Linear Algebra Xiao Yu Wang MIT Aug 2010 Xiao Yu Wang (MIT) Math Camp 2010 08/10 1 / 88 Linear Algebra Game Plan Vector Spaces Linear Transformations and Matrices Determinant
More informationMA 527 first midterm review problems Hopefully final version as of October 2nd
MA 57 first midterm review problems Hopefully final version as of October nd The first midterm will be on Wednesday, October 4th, from 8 to 9 pm, in MTHW 0. It will cover all the material from the classes
More informationContents. 1 State-Space Linear Systems 5. 2 Linearization Causality, Time Invariance, and Linearity 31
Contents Preamble xiii Linear Systems I Basic Concepts 1 I System Representation 3 1 State-Space Linear Systems 5 1.1 State-Space Linear Systems 5 1.2 Block Diagrams 7 1.3 Exercises 11 2 Linearization
More informationFull-State Feedback Design for a Multi-Input System
Full-State Feedback Design for a Multi-Input System A. Introduction The open-loop system is described by the following state space model. x(t) = Ax(t)+Bu(t), y(t) =Cx(t)+Du(t) () 4 8.5 A =, B =.5.5, C
More informationOutline. Linear Algebra for Computer Vision
Outline Linear Algebra for Computer Vision Introduction CMSC 88 D Notation and Basics Motivation Linear systems of equations Gauss Elimination, LU decomposition Linear Spaces and Operators Addition, scalar
More informationECS231: Spectral Partitioning. Based on Berkeley s CS267 lecture on graph partition
ECS231: Spectral Partitioning Based on Berkeley s CS267 lecture on graph partition 1 Definition of graph partitioning Given a graph G = (N, E, W N, W E ) N = nodes (or vertices), E = edges W N = node weights
More informationJoão P. Hespanha. January 16, 2009
LINEAR SYSTEMS THEORY João P. Hespanha January 16, 2009 Disclaimer: This is a draft and probably contains a few typos. Comments and information about typos are welcome. Please contact the author at hespanha@ece.ucsb.edu.
More informationMATH 583A REVIEW SESSION #1
MATH 583A REVIEW SESSION #1 BOJAN DURICKOVIC 1. Vector Spaces Very quick review of the basic linear algebra concepts (see any linear algebra textbook): (finite dimensional) vector space (or linear space),
More informationAn Introduction to Spectral Graph Theory
An Introduction to Spectral Graph Theory Mackenzie Wheeler Supervisor: Dr. Gary MacGillivray University of Victoria WheelerM@uvic.ca Outline Outline 1. How many walks are there from vertices v i to v j
More informationMidterm for Introduction to Numerical Analysis I, AMSC/CMSC 466, on 10/29/2015
Midterm for Introduction to Numerical Analysis I, AMSC/CMSC 466, on 10/29/2015 The test lasts 1 hour and 15 minutes. No documents are allowed. The use of a calculator, cell phone or other equivalent electronic
More informationNetworks and Their Spectra
Networks and Their Spectra Victor Amelkin University of California, Santa Barbara Department of Computer Science victor@cs.ucsb.edu December 4, 2017 1 / 18 Introduction Networks (= graphs) are everywhere.
More informationLecture 12 : Graph Laplacians and Cheeger s Inequality
CPS290: Algorithmic Foundations of Data Science March 7, 2017 Lecture 12 : Graph Laplacians and Cheeger s Inequality Lecturer: Kamesh Munagala Scribe: Kamesh Munagala Graph Laplacian Maybe the most beautiful
More informationLINEAR ALGEBRA 1, 2012-I PARTIAL EXAM 3 SOLUTIONS TO PRACTICE PROBLEMS
LINEAR ALGEBRA, -I PARTIAL EXAM SOLUTIONS TO PRACTICE PROBLEMS Problem (a) For each of the two matrices below, (i) determine whether it is diagonalizable, (ii) determine whether it is orthogonally diagonalizable,
More informationDIAGONALIZATION OF THE STRESS TENSOR
DIAGONALIZATION OF THE STRESS TENSOR INTRODUCTION By the use of Cauchy s theorem we are able to reduce the number of stress components in the stress tensor to only nine values. An additional simplification
More informationMath Final December 2006 C. Robinson
Math 285-1 Final December 2006 C. Robinson 2 5 8 5 1 2 0-1 0 1. (21 Points) The matrix A = 1 2 2 3 1 8 3 2 6 has the reduced echelon form U = 0 0 1 2 0 0 0 0 0 1. 2 6 1 0 0 0 0 0 a. Find a basis for the
More informationI. Multiple Choice Questions (Answer any eight)
Name of the student : Roll No : CS65: Linear Algebra and Random Processes Exam - Course Instructor : Prashanth L.A. Date : Sep-24, 27 Duration : 5 minutes INSTRUCTIONS: The test will be evaluated ONLY
More informationLecture 13 Spectral Graph Algorithms
COMS 995-3: Advanced Algorithms March 6, 7 Lecture 3 Spectral Graph Algorithms Instructor: Alex Andoni Scribe: Srikar Varadaraj Introduction Today s topics: Finish proof from last lecture Example of random
More information5.) For each of the given sets of vectors, determine whether or not the set spans R 3. Give reasons for your answers.
Linear Algebra - Test File - Spring Test # For problems - consider the following system of equations. x + y - z = x + y + 4z = x + y + 6z =.) Solve the system without using your calculator..) Find the
More informationLecture 9: Laplacian Eigenmaps
Lecture 9: Radu Balan Department of Mathematics, AMSC, CSCAMM and NWC University of Maryland, College Park, MD April 18, 2017 Optimization Criteria Assume G = (V, W ) is a undirected weighted graph with
More informationThe spectra of super line multigraphs
The spectra of super line multigraphs Jay Bagga Department of Computer Science Ball State University Muncie, IN jbagga@bsuedu Robert B Ellis Department of Applied Mathematics Illinois Institute of Technology
More informationITTC Science of Communication Networks The University of Kansas EECS SCN Graph Spectra
Science of Communication Networks The University of Kansas EECS SCN Graph Spectra Egemen K. Çetinkaya and James P.G. Sterbenz Department of Electrical Engineering & Computer Science Information Technology
More informationMethods for sparse analysis of high-dimensional data, II
Methods for sparse analysis of high-dimensional data, II Rachel Ward May 26, 2011 High dimensional data with low-dimensional structure 300 by 300 pixel images = 90, 000 dimensions 2 / 55 High dimensional
More informationExample Linear Algebra Competency Test
Example Linear Algebra Competency Test The 4 questions below are a combination of True or False, multiple choice, fill in the blank, and computations involving matrices and vectors. In the latter case,
More informationMatrix Vector Products
We covered these notes in the tutorial sessions I strongly recommend that you further read the presented materials in classical books on linear algebra Please make sure that you understand the proofs and
More informationSpectral radius, symmetric and positive matrices
Spectral radius, symmetric and positive matrices Zdeněk Dvořák April 28, 2016 1 Spectral radius Definition 1. The spectral radius of a square matrix A is ρ(a) = max{ λ : λ is an eigenvalue of A}. For an
More informationConditioning of the Entries in the Stationary Vector of a Google-Type Matrix. Steve Kirkland University of Regina
Conditioning of the Entries in the Stationary Vector of a Google-Type Matrix Steve Kirkland University of Regina June 5, 2006 Motivation: Google s PageRank algorithm finds the stationary vector of a stochastic
More informationA note on the minimal volume of almost cubic parallelepipeds
A note on the minimal volume of almost cubic parallelepipeds Daniele Micciancio Abstract We prove that the best way to reduce the volume of the n-dimensional unit cube by a linear transformation that maps
More informationMULTI-AGENT TRACKING OF A HIGH-DIMENSIONAL ACTIVE LEADER WITH SWITCHING TOPOLOGY
Jrl Syst Sci & Complexity (2009) 22: 722 731 MULTI-AGENT TRACKING OF A HIGH-DIMENSIONAL ACTIVE LEADER WITH SWITCHING TOPOLOGY Yiguang HONG Xiaoli WANG Received: 11 May 2009 / Revised: 16 June 2009 c 2009
More informationNumerical Solutions to PDE s
Introduction Numerical Solutions to PDE s Mathematical Modelling Week 5 Kurt Bryan Let s start by recalling a simple numerical scheme for solving ODE s. Suppose we have an ODE u (t) = f(t, u(t)) for some
More informationAlgebraic Multigrid Preconditioners for Computing Stationary Distributions of Markov Processes
Algebraic Multigrid Preconditioners for Computing Stationary Distributions of Markov Processes Elena Virnik, TU BERLIN Algebraic Multigrid Preconditioners for Computing Stationary Distributions of Markov
More information33AH, WINTER 2018: STUDY GUIDE FOR FINAL EXAM
33AH, WINTER 2018: STUDY GUIDE FOR FINAL EXAM (UPDATED MARCH 17, 2018) The final exam will be cumulative, with a bit more weight on more recent material. This outline covers the what we ve done since the
More informationMATH 1553 PRACTICE FINAL EXAMINATION
MATH 553 PRACTICE FINAL EXAMINATION Name Section 2 3 4 5 6 7 8 9 0 Total Please read all instructions carefully before beginning. The final exam is cumulative, covering all sections and topics on the master
More informationStability Analysis of Stochastically Varying Formations of Dynamic Agents
Stability Analysis of Stochastically Varying Formations of Dynamic Agents Vijay Gupta, Babak Hassibi and Richard M. Murray Division of Engineering and Applied Science California Institute of Technology
More informationLaplacians of Graphs, Spectra and Laplacian polynomials
Laplacians of Graphs, Spectra and Laplacian polynomials Lector: Alexander Mednykh Sobolev Institute of Mathematics Novosibirsk State University Winter School in Harmonic Functions on Graphs and Combinatorial
More information1. What is the determinant of the following matrix? a 1 a 2 4a 3 2a 2 b 1 b 2 4b 3 2b c 1. = 4, then det
What is the determinant of the following matrix? 3 4 3 4 3 4 4 3 A 0 B 8 C 55 D 0 E 60 If det a a a 3 b b b 3 c c c 3 = 4, then det a a 4a 3 a b b 4b 3 b c c c 3 c = A 8 B 6 C 4 D E 3 Let A be an n n matrix
More informationMeasurement partitioning and observational. equivalence in state estimation
Measurement partitioning and observational 1 equivalence in state estimation Mohammadreza Doostmohammadian, Student Member, IEEE, and Usman A. Khan, Senior Member, IEEE arxiv:1412.5111v1 [cs.it] 16 Dec
More information