Lecture Note 1: Background
|
|
- Mark Austin
- 6 years ago
- Views:
Transcription
1 ECE5463: Introduction to Robotics Lecture Note 1: Background Prof. Wei Zhang Department of Electrical and Computer Engineering Ohio State University Columbus, Ohio, USA Spring 2018 Lecture 1 (ECE5463 Sp18) Wei Zhang(OSU) 1 / 21
2 Outline Linear Algebra Linear Differential Equation Linear and Angular Motion of Point Mass Outline Lecture 1 (ECE5463 Sp18) Wei Zhang(OSU) 2 / 21
3 Free Vector Free Vector: geometric quantify with length and direction free means not necessarily rooted anywhere; only length and direction matter Given a reference frame, v can be moved to a position such that the base of the arrow is at the origin without changing the orientation. Then the vector v can be represented its coordinates v in the reference frame. v denotes the physical quantify while v denote its coordinate wrt some frame. Linear Algebra Lecture 1 (ECE5463 Sp18) Wei Zhang(OSU) 3 / 21
4 Point and Its Coordinate Point: p denotes a point in the physical space A point p can be represented by as a vector from frame origin to p p denotes the coordinate of a point p The coordinate p depends on the choice of reference frame Linear Algebra Lecture 1 (ECE5463 Sp18) Wei Zhang(OSU) 4 / 21
5 Vector (Math) Vector: p R n : Inner product of two vectors p R n, q R n : Norm of a vector p: Angle between two vectors p, q R n : Linear Algebra Lecture 1 (ECE5463 Sp18) Wei Zhang(OSU) 5 / 21
6 Matrix A R n m Symmetric matrix Matrix vector multiplication as linear combination of columns Linear Algebra Lecture 1 (ECE5463 Sp18) Wei Zhang(OSU) 6 / 21
7 Change of Basis Two Bases for R n : {a} ={â 1,..., â n } and {b} = {ˆb 1,..., ˆb n } v a and v b are the corresponding coordinates of v w.r.t. {a} and {b}, how to they relate? Linear Algebra Lecture 1 (ECE5463 Sp18) Wei Zhang(OSU) 7 / 21
8 Cross Product Cross product or vector product of a R 3, b R 3 is defined as a b = a 2b 3 a 3 b 2 a 3 b 1 a 1 b 3 (1) a 1 b 2 a 2 b 1 Properties: a b = a b sin(θ) a b = b a a a = 0 Linear Algebra Lecture 1 (ECE5463 Sp18) Wei Zhang(OSU) 8 / 21
9 Skew symmetric representation It can be directly verified from definition that a b = [a]b, where 0 a 3 a 2 [a] a 3 0 a 1 (2) a 2 a 1 0 a = a 1 a 2 a 3 [a] [a] = [a] T (called skew symmetric) Example: Linear Algebra Lecture 1 (ECE5463 Sp18) Wei Zhang(OSU) 9 / 21
10 Positive Semidefinite Matrix A symmetric square matrix A R n n is called positive semidefinite (p.s.d.), denoted by A 0, if x T Ax 0, x R n A symmetric square matrix A R n n is called positive definite (p.d.), denoted by A 0, if x T Ax > 0 for all nonzero x R n p.d. matrices characterize positive definite quadratic forms: Linear Algebra Lecture 1 (ECE5463 Sp18) Wei Zhang(OSU) 10 / 21
11 Positive Semidefinite Matrix II Equivalent definitions for p.s.d. matrices: - All eigs of A are nonnegative - There exists a factorization A = B T B Equivalent definitions for p.d. matrices: - All eigs of A are strictly positive - There exists a factorization A = B T B with B square and nonsingular. If A R n n is p.d., then A 1 is also p.d. Linear Algebra Lecture 1 (ECE5463 Sp18) Wei Zhang(OSU) 11 / 21
12 Ellipsoid in R n Unit sphere in R n : S = {x R n : x x c = 1} Ellipsoid in R n : S = {x R n : (x x c ) T A 1 (x x c ) = 1}, for some p.d. A R n n. Let λ 1,..., λ n be the eigenvalues of A with corresponding eigenvectors v 1,..., v n. - Principal semi-axis lengths are λ 1,..., λ n - Direction of Principal semi-axes are aligned with v 1,..., v n - volume of the ellipsoid is proportional to det(a) Linear Algebra Lecture 1 (ECE5463 Sp18) Wei Zhang(OSU) 12 / 21
13 Outline Linear Algebra Linear Differential Equation Linear and Angular Motion of Point Mass Linear ODE Lecture 1 (ECE5463 Sp18) Wei Zhang(OSU) 13 / 21
14 Scalar Linear Differential Equation ẋ(t) = ax(t), with initial condition x(0) = x 0 (3) x(t) R, a R is constant The above ODE has a unique solution x(t) = e at x 0 What is the number e? Linear ODE Lecture 1 (ECE5463 Sp18) Wei Zhang(OSU) 14 / 21
15 Complex Exponential For real variable x R, Taylor series expansion for e x around x = 0: e x = k=0 This can be extended to complex variables: e z = k=0 x k k! = 1 + x + x2 2! + x3 3! + z k k! = 1 + z + z2 2! + z3 3! + This power series is well defined for all z C In particular, we have e jθ = 1 + jθ θ2 2 j θ3 3! + Comparing with Taylor expansions for cos(θ) and sin(θ) leads to the Euler Identity Linear ODE Lecture 1 (ECE5463 Sp18) Wei Zhang(OSU) 15 / 21
16 Matrix Exponential Similar to the real and complex cases, we can define the so-called matrix exponential e A A k = I + A + A2 k! 2! + A3 3! + k=0 This power series is well defined whenever A is finite and constant. One can verify directly from definition: - Ae A = e A A - If A = P DP 1, then e A = P e D P 1 Linear ODE Lecture 1 (ECE5463 Sp18) Wei Zhang(OSU) 16 / 21
17 Vector Linear Differential Equation ẋ(t) = Ax(t), with initial condition x(0) = x 0 (4) x(t) R n, A R n n is constant matrix, x 0 R n is given. With the definition of matrix exponential, we can show that the solution to (4) is given by x(t) = e At x 0. Linear ODE Lecture 1 (ECE5463 Sp18) Wei Zhang(OSU) 17 / 21
18 Example Find the solution to ẋ(t) = [ ] x(t) Linear ODE Lecture 1 (ECE5463 Sp18) Wei Zhang(OSU) 18 / 21
19 Outline Linear Algebra Linear Differential Equation Linear and Angular Motion of Point Mass Point Mass Motion Lecture 1 (ECE5463 Sp18) Wei Zhang(OSU) 19 / 21
20 Linear Motion Consider a particle with mass m position velocity/acceleration Force Momentum Newton s Second Law Point Mass Motion Lecture 1 (ECE5463 Sp18) Wei Zhang(OSU) 20 / 21
21 Angular Motion Angle Angular velocity Torque (Moment) Angular Momentum Newton s Second Law Point Mass Motion Lecture 1 (ECE5463 Sp18) Wei Zhang(OSU) 21 / 21
Lecture Note 5: Semidefinite Programming for Stability Analysis
ECE7850: Hybrid Systems:Theory and Applications Lecture Note 5: Semidefinite Programming for Stability Analysis Wei Zhang Assistant Professor Department of Electrical and Computer Engineering Ohio State
More informationLecture Note 8: Inverse Kinematics
ECE5463: Introduction to Robotics Lecture Note 8: Inverse Kinematics Prof. Wei Zhang Department of Electrical and Computer Engineering Ohio State University Columbus, Ohio, USA Spring 2018 Lecture 8 (ECE5463
More informationLecture Note 8: Inverse Kinematics
ECE5463: Introduction to Robotics Lecture Note 8: Inverse Kinematics Prof. Wei Zhang Department of Electrical and Computer Engineering Ohio State University Columbus, Ohio, USA Spring 2018 Lecture 8 (ECE5463
More informationProperties of Linear Transformations from R n to R m
Properties of Linear Transformations from R n to R m MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Topic Overview Relationship between the properties of a matrix transformation
More informationMath Matrix Algebra
Math 44 - Matrix Algebra Review notes - (Alberto Bressan, Spring 7) sec: Orthogonal diagonalization of symmetric matrices When we seek to diagonalize a general n n matrix A, two difficulties may arise:
More informationLecture Note 7: Velocity Kinematics and Jacobian
ECE5463: Introduction to Robotics Lecture Note 7: Velocity Kinematics and Jacobian Prof. Wei Zhang Department of Electrical and Computer Engineering Ohio State University Columbus, Ohio, USA Spring 2018
More informationIntroduction to Matrix Algebra
Introduction to Matrix Algebra August 18, 2010 1 Vectors 1.1 Notations A p-dimensional vector is p numbers put together. Written as x 1 x =. x p. When p = 1, this represents a point in the line. When p
More informationLecture Note 4: General Rigid Body Motion
ECE5463: Introduction to Robotics Lecture Note 4: General Rigid Body Motion Prof. Wei Zhang Department of Electrical and Computer Engineering Ohio State University Columbus, Ohio, USA Spring 2018 Lecture
More informationLecture Note 5: Velocity of a Rigid Body
ECE5463: Introduction to Robotics Lecture Note 5: Velocity of a Rigid Body Prof. Wei Zhang Department of Electrical and Computer Engineering Ohio State University Columbus, Ohio, USA Spring 2018 Lecture
More informationChapter 3 Transformations
Chapter 3 Transformations An Introduction to Optimization Spring, 2014 Wei-Ta Chu 1 Linear Transformations A function is called a linear transformation if 1. for every and 2. for every If we fix the bases
More informationPrincipal Components Theory Notes
Principal Components Theory Notes Charles J. Geyer August 29, 2007 1 Introduction These are class notes for Stat 5601 (nonparametrics) taught at the University of Minnesota, Spring 2006. This not a theory
More informationLinear Algebra: Matrix Eigenvalue Problems
CHAPTER8 Linear Algebra: Matrix Eigenvalue Problems Chapter 8 p1 A matrix eigenvalue problem considers the vector equation (1) Ax = λx. 8.0 Linear Algebra: Matrix Eigenvalue Problems Here A is a given
More informationMATH 304 Linear Algebra Lecture 20: The Gram-Schmidt process (continued). Eigenvalues and eigenvectors.
MATH 304 Linear Algebra Lecture 20: The Gram-Schmidt process (continued). Eigenvalues and eigenvectors. Orthogonal sets Let V be a vector space with an inner product. Definition. Nonzero vectors v 1,v
More informationLecture Note 12: Dynamics of Open Chains: Lagrangian Formulation
ECE5463: Introduction to Robotics Lecture Note 12: Dynamics of Open Chains: Lagrangian Formulation Prof. Wei Zhang Department of Electrical and Computer Engineering Ohio State University Columbus, Ohio,
More informationLecture Note 7: Velocity Kinematics and Jacobian
ECE5463: Introduction to Robotics Lecture Note 7: Velocity Kinematics and Jacobian Prof. Wei Zhang Department of Electrical and Computer Engineering Ohio State University Columbus, Ohio, USA Spring 2018
More informationLecture 6 Positive Definite Matrices
Linear Algebra Lecture 6 Positive Definite Matrices Prof. Chun-Hung Liu Dept. of Electrical and Computer Engineering National Chiao Tung University Spring 2017 2017/6/8 Lecture 6: Positive Definite Matrices
More informationWeek Quadratic forms. Principal axes theorem. Text reference: this material corresponds to parts of sections 5.5, 8.2,
Math 051 W008 Margo Kondratieva Week 10-11 Quadratic forms Principal axes theorem Text reference: this material corresponds to parts of sections 55, 8, 83 89 Section 41 Motivation and introduction Consider
More informationReview: control, feedback, etc. Today s topic: state-space models of systems; linearization
Plan of the Lecture Review: control, feedback, etc Today s topic: state-space models of systems; linearization Goal: a general framework that encompasses all examples of interest Once we have mastered
More informationMath Bootcamp An p-dimensional vector is p numbers put together. Written as. x 1 x =. x p
Math Bootcamp 2012 1 Review of matrix algebra 1.1 Vectors and rules of operations An p-dimensional vector is p numbers put together. Written as x 1 x =. x p. When p = 1, this represents a point in the
More informationLinear Algebra - Part II
Linear Algebra - Part II Projection, Eigendecomposition, SVD (Adapted from Sargur Srihari s slides) Brief Review from Part 1 Symmetric Matrix: A = A T Orthogonal Matrix: A T A = AA T = I and A 1 = A T
More informationRobotics & Automation. Lecture 03. Representation of SO(3) John T. Wen. September 3, 2008
Robotics & Automation Lecture 03 Representation of SO(3) John T. Wen September 3, 2008 Last Time Transformation of vectors: v a = R ab v b Transformation of linear transforms: L a = R ab L b R ba R SO(3)
More informationRecall the convention that, for us, all vectors are column vectors.
Some linear algebra Recall the convention that, for us, all vectors are column vectors. 1. Symmetric matrices Let A be a real matrix. Recall that a complex number λ is an eigenvalue of A if there exists
More informationLecture 3: Review of Linear Algebra
ECE 83 Fall 2 Statistical Signal Processing instructor: R Nowak Lecture 3: Review of Linear Algebra Very often in this course we will represent signals as vectors and operators (eg, filters, transforms,
More informationLecture 3: Review of Linear Algebra
ECE 83 Fall 2 Statistical Signal Processing instructor: R Nowak, scribe: R Nowak Lecture 3: Review of Linear Algebra Very often in this course we will represent signals as vectors and operators (eg, filters,
More information1 Overview. 2 A Characterization of Convex Functions. 2.1 First-order Taylor approximation. AM 221: Advanced Optimization Spring 2016
AM 221: Advanced Optimization Spring 2016 Prof. Yaron Singer Lecture 8 February 22nd 1 Overview In the previous lecture we saw characterizations of optimality in linear optimization, and we reviewed the
More informationIn these chapter 2A notes write vectors in boldface to reduce the ambiguity of the notation.
1 2 Linear Systems In these chapter 2A notes write vectors in boldface to reduce the ambiguity of the notation 21 Matrix ODEs Let and is a scalar A linear function satisfies Linear superposition ) Linear
More informationChap 3. Linear Algebra
Chap 3. Linear Algebra Outlines 1. Introduction 2. Basis, Representation, and Orthonormalization 3. Linear Algebraic Equations 4. Similarity Transformation 5. Diagonal Form and Jordan Form 6. Functions
More informationLecture 02 Linear Algebra Basics
Introduction to Computational Data Analysis CX4240, 2019 Spring Lecture 02 Linear Algebra Basics Chao Zhang College of Computing Georgia Tech These slides are based on slides from Le Song and Andres Mendez-Vazquez.
More informationMay 9, 2014 MATH 408 MIDTERM EXAM OUTLINE. Sample Questions
May 9, 24 MATH 48 MIDTERM EXAM OUTLINE This exam will consist of two parts and each part will have multipart questions. Each of the 6 questions is worth 5 points for a total of points. The two part of
More information21 Linear State-Space Representations
ME 132, Spring 25, UC Berkeley, A Packard 187 21 Linear State-Space Representations First, let s describe the most general type of dynamic system that we will consider/encounter in this class Systems may
More informationMatrices and Deformation
ES 111 Mathematical Methods in the Earth Sciences Matrices and Deformation Lecture Outline 13 - Thurs 9th Nov 2017 Strain Ellipse and Eigenvectors One way of thinking about a matrix is that it operates
More informationChapter Two Elements of Linear Algebra
Chapter Two Elements of Linear Algebra Previously, in chapter one, we have considered single first order differential equations involving a single unknown function. In the next chapter we will begin to
More informationLecture 38: Equations of Rigid-Body Motion
Lecture 38: Equations of Rigid-Body Motion It s going to be easiest to find the equations of motion for the object in the body frame i.e., the frame where the axes are principal axes In general, we can
More information2. Review of Linear Algebra
2. Review of Linear Algebra ECE 83, Spring 217 In this course we will represent signals as vectors and operators (e.g., filters, transforms, etc) as matrices. This lecture reviews basic concepts from linear
More informationMath 302 Outcome Statements Winter 2013
Math 302 Outcome Statements Winter 2013 1 Rectangular Space Coordinates; Vectors in the Three-Dimensional Space (a) Cartesian coordinates of a point (b) sphere (c) symmetry about a point, a line, and a
More informationLecture 7. Econ August 18
Lecture 7 Econ 2001 2015 August 18 Lecture 7 Outline First, the theorem of the maximum, an amazing result about continuity in optimization problems. Then, we start linear algebra, mostly looking at familiar
More informationMATH.2720 Introduction to Programming with MATLAB Vector and Matrix Algebra
MATH.2720 Introduction to Programming with MATLAB Vector and Matrix Algebra A. Vectors A vector is a quantity that has both magnitude and direction, like velocity. The location of a vector is irrelevant;
More informationConvex Functions. Daniel P. Palomar. Hong Kong University of Science and Technology (HKUST)
Convex Functions Daniel P. Palomar Hong Kong University of Science and Technology (HKUST) ELEC5470 - Convex Optimization Fall 2017-18, HKUST, Hong Kong Outline of Lecture Definition convex function Examples
More informationComputational math: Assignment 1
Computational math: Assignment 1 Thanks Ting Gao for her Latex file 11 Let B be a 4 4 matrix to which we apply the following operations: 1double column 1, halve row 3, 3add row 3 to row 1, 4interchange
More informationGlobal Maxwellians over All Space and Their Relation to Conserved Quantites of Classical Kinetic Equations
Global Maxwellians over All Space and Their Relation to Conserved Quantites of Classical Kinetic Equations C. David Levermore Department of Mathematics and Institute for Physical Science and Technology
More informationMath 102, Winter Final Exam Review. Chapter 1. Matrices and Gaussian Elimination
Math 0, Winter 07 Final Exam Review Chapter. Matrices and Gaussian Elimination { x + x =,. Different forms of a system of linear equations. Example: The x + 4x = 4. [ ] [ ] [ ] vector form (or the column
More informationLinear Algebra And Its Applications Chapter 6. Positive Definite Matrix
Linear Algebra And Its Applications Chapter 6. Positive Definite Matrix KAIST wit Lab 2012. 07. 10 남성호 Introduction The signs of the eigenvalues can be important. The signs can also be related to the minima,
More informationLecture 38: Equations of Rigid-Body Motion
Lecture 38: Equations of Rigid-Body Motion It s going to be easiest to find the equations of motion for the object in the body frame i.e., the frame where the axes are principal axes In general, we can
More informationft-uiowa-math2550 Assignment NOTRequiredJustHWformatOfQuizReviewForExam3part2 due 12/31/2014 at 07:10pm CST
me me ft-uiowa-math2550 Assignment NOTRequiredJustHWformatOfQuizReviewForExam3part2 due 12/31/2014 at 07:10pm CST 1. (1 pt) local/library/ui/eigentf.pg A is n n an matrices.. There are an infinite number
More informationSection 13.4 The Cross Product
Section 13.4 The Cross Product Multiplying Vectors 2 In this section we consider the more technical multiplication which can be defined on vectors in 3-space (but not vectors in 2-space). 1. Basic Definitions
More informationFINAL EXAM MATH303 Theory of Ordinary Differential Equations. Spring dx dt = x + 3y dy dt = x y.
FINAL EXAM MATH0 Theory of Ordinary Differential Equations There are 5 problems on 2 pages. Spring 2009. 25 points Consider the linear plane autonomous system x + y x y. Find a fundamental matrix of the
More informationLecture 37: Principal Axes, Translations, and Eulerian Angles
Lecture 37: Principal Axes, Translations, and Eulerian Angles When Can We Find Principal Axes? We can always write down the cubic equation that one must solve to determine the principal moments But if
More informationConjugate Gradient (CG) Method
Conjugate Gradient (CG) Method by K. Ozawa 1 Introduction In the series of this lecture, I will introduce the conjugate gradient method, which solves efficiently large scale sparse linear simultaneous
More informationx. Figure 1: Examples of univariate Gaussian pdfs N (x; µ, σ 2 ).
.8.6 µ =, σ = 1 µ = 1, σ = 1 / µ =, σ =.. 3 1 1 3 x Figure 1: Examples of univariate Gaussian pdfs N (x; µ, σ ). The Gaussian distribution Probably the most-important distribution in all of statistics
More information2. Matrix Algebra and Random Vectors
2. Matrix Algebra and Random Vectors 2.1 Introduction Multivariate data can be conveniently display as array of numbers. In general, a rectangular array of numbers with, for instance, n rows and p columns
More informationRobot Control Basics CS 685
Robot Control Basics CS 685 Control basics Use some concepts from control theory to understand and learn how to control robots Control Theory general field studies control and understanding of behavior
More informationMath 241, Exam 1 Information.
Math 241, Exam 1 Information. 2/13/13, LC 310, 11:15-12:05. Exam 1 will be based on: Sections 12.1-12.5, 14.2. The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/241sp13/241.html)
More informationVector Spaces and SubSpaces
Vector Spaces and SubSpaces Linear Algebra MATH 2076 Linear Algebra Vector Spaces & SubSpaces Chapter 4, Section 1b 1 / 10 What is a Vector Space? A vector space is a bunch of objects that we call vectors
More informationSome New Results on Lyapunov-Type Diagonal Stability
Some New Results on Lyapunov-Type Diagonal Stability Mehmet Gumus (Joint work with Dr. Jianhong Xu) Department of Mathematics Southern Illinois University Carbondale 12/01/2016 mgumus@siu.edu (SIUC) Lyapunov-Type
More informationEXERCISES ON DETERMINANTS, EIGENVALUES AND EIGENVECTORS. 1. Determinants
EXERCISES ON DETERMINANTS, EIGENVALUES AND EIGENVECTORS. Determinants Ex... Let A = 0 4 4 2 0 and B = 0 3 0. (a) Compute 0 0 0 0 A. (b) Compute det(2a 2 B), det(4a + B), det(2(a 3 B 2 )). 0 t Ex..2. For
More informationMATH 1553, C. JANKOWSKI MIDTERM 3
MATH 1553, C JANKOWSKI MIDTERM 3 Name GT Email @gatechedu Write your section number (E6-E9) here: Please read all instructions carefully before beginning Please leave your GT ID card on your desk until
More informationMath 265 Linear Algebra Sample Spring 2002., rref (A) =
Math 265 Linear Algebra Sample Spring 22. It is given that A = rref (A T )= 2 3 5 3 2 6, rref (A) = 2 3 and (a) Find the rank of A. (b) Find the nullityof A. (c) Find a basis for the column space of A.
More informationSpan and Linear Independence
Span and Linear Independence It is common to confuse span and linear independence, because although they are different concepts, they are related. To see their relationship, let s revisit the previous
More informationHomework sheet 4: EIGENVALUES AND EIGENVECTORS. DIAGONALIZATION (with solutions) Year ? Why or why not? 6 9
Bachelor in Statistics and Business Universidad Carlos III de Madrid Mathematical Methods II María Barbero Liñán Homework sheet 4: EIGENVALUES AND EIGENVECTORS DIAGONALIZATION (with solutions) Year - Is
More informationMATH 1553 PRACTICE MIDTERM 3 (VERSION B)
MATH 1553 PRACTICE MIDTERM 3 (VERSION B) Name Section 1 2 3 4 5 Total Please read all instructions carefully before beginning. Each problem is worth 10 points. The maximum score on this exam is 50 points.
More informationMATH 213 Linear Algebra and Ordinary Differential Equations Spring 2015 Study Sheet for Final Exam. Topics
MATH 213 Linear Algebra and Ordinary Differential Equations Spring 2015 Study Sheet for Final Exam This study sheet will not be allowed during the test. Books and notes will not be allowed during the test.
More informationLinear Algebra review Powers of a diagonalizable matrix Spectral decomposition
Linear Algebra review Powers of a diagonalizable matrix Spectral decomposition Prof. Tesler Math 283 Fall 2016 Also see the separate version of this with Matlab and R commands. Prof. Tesler Diagonalizing
More informationReview of Some Concepts from Linear Algebra: Part 2
Review of Some Concepts from Linear Algebra: Part 2 Department of Mathematics Boise State University January 16, 2019 Math 566 Linear Algebra Review: Part 2 January 16, 2019 1 / 22 Vector spaces A set
More informationMath 315: Linear Algebra Solutions to Assignment 7
Math 5: Linear Algebra s to Assignment 7 # Find the eigenvalues of the following matrices. (a.) 4 0 0 0 (b.) 0 0 9 5 4. (a.) The characteristic polynomial det(λi A) = (λ )(λ )(λ ), so the eigenvalues are
More informationPosition and orientation of rigid bodies
Robotics 1 Position and orientation of rigid bodies Prof. Alessandro De Luca Robotics 1 1 Position and orientation right-handed orthogonal Reference Frames RF A A z A p AB B RF B z B x B y A rigid body
More informationHomework 2 Foundations of Computational Math 2 Spring 2019
Homework 2 Foundations of Computational Math 2 Spring 2019 Problem 2.1 (2.1.a) Suppose (v 1,λ 1 )and(v 2,λ 2 ) are eigenpairs for a matrix A C n n. Show that if λ 1 λ 2 then v 1 and v 2 are linearly independent.
More information1. Matrix multiplication and Pauli Matrices: Pauli matrices are the 2 2 matrices. 1 0 i 0. 0 i
Problems in basic linear algebra Science Academies Lecture Workshop at PSGRK College Coimbatore, June 22-24, 2016 Govind S. Krishnaswami, Chennai Mathematical Institute http://www.cmi.ac.in/~govind/teaching,
More informationRecall : Eigenvalues and Eigenvectors
Recall : Eigenvalues and Eigenvectors Let A be an n n matrix. If a nonzero vector x in R n satisfies Ax λx for a scalar λ, then : The scalar λ is called an eigenvalue of A. The vector x is called an eigenvector
More informationMath Matrix Algebra
Math 44 - Matrix Algebra Review notes - 4 (Alberto Bressan, Spring 27) Review of complex numbers In this chapter we shall need to work with complex numbers z C These can be written in the form z = a+ib,
More informationIntroduction and Math Preliminaries
Introduction and Math Preliminaries Yinyu Ye Department of Management Science and Engineering Stanford University Stanford, CA 94305, U.S.A. http://www.stanford.edu/ yyye Appendices A, B, and C, Chapter
More informationSymmetric and anti symmetric matrices
Symmetric and anti symmetric matrices In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, matrix A is symmetric if. A = A Because equal matrices have equal
More informationMATH 304 Linear Algebra Lecture 34: Review for Test 2.
MATH 304 Linear Algebra Lecture 34: Review for Test 2. Topics for Test 2 Linear transformations (Leon 4.1 4.3) Matrix transformations Matrix of a linear mapping Similar matrices Orthogonality (Leon 5.1
More informationMCE/EEC 647/747: Robot Dynamics and Control. Lecture 8: Basic Lyapunov Stability Theory
MCE/EEC 647/747: Robot Dynamics and Control Lecture 8: Basic Lyapunov Stability Theory Reading: SHV Appendix Mechanical Engineering Hanz Richter, PhD MCE503 p.1/17 Stability in the sense of Lyapunov A
More information1. Select the unique answer (choice) for each problem. Write only the answer.
MATH 5 Practice Problem Set Spring 7. Select the unique answer (choice) for each problem. Write only the answer. () Determine all the values of a for which the system has infinitely many solutions: x +
More informationCAAM 335 Matrix Analysis
CAAM 335 Matrix Analysis Solutions to Homework 8 Problem (5+5+5=5 points The partial fraction expansion of the resolvent for the matrix B = is given by (si B = s } {{ } =P + s + } {{ } =P + (s (5 points
More informationMTH 464: Computational Linear Algebra
MTH 464: Computational Linear Algebra Lecture Outlines Exam 4 Material Prof. M. Beauregard Department of Mathematics & Statistics Stephen F. Austin State University April 15, 2018 Linear Algebra (MTH 464)
More informationMultiple Degree of Freedom Systems. The Millennium bridge required many degrees of freedom to model and design with.
Multiple Degree of Freedom Systems The Millennium bridge required many degrees of freedom to model and design with. The first step in analyzing multiple degrees of freedom (DOF) is to look at DOF DOF:
More informationNumerical Methods for ODEs. Lectures for PSU Summer Programs Xiantao Li
Numerical Methods for ODEs Lectures for PSU Summer Programs Xiantao Li Outline Introduction Some Challenges Numerical methods for ODEs Stiff ODEs Accuracy Constrained dynamics Stability Coarse-graining
More informationBasic Concepts in Matrix Algebra
Basic Concepts in Matrix Algebra An column array of p elements is called a vector of dimension p and is written as x p 1 = x 1 x 2. x p. The transpose of the column vector x p 1 is row vector x = [x 1
More informationLecture 7: Positive Semidefinite Matrices
Lecture 7: Positive Semidefinite Matrices Rajat Mittal IIT Kanpur The main aim of this lecture note is to prepare your background for semidefinite programming. We have already seen some linear algebra.
More informationLecture 10 - Eigenvalues problem
Lecture 10 - Eigenvalues problem Department of Computer Science University of Houston February 28, 2008 1 Lecture 10 - Eigenvalues problem Introduction Eigenvalue problems form an important class of problems
More informationTMA Calculus 3. Lecture 21, April 3. Toke Meier Carlsen Norwegian University of Science and Technology Spring 2013
TMA4115 - Calculus 3 Lecture 21, April 3 Toke Meier Carlsen Norwegian University of Science and Technology Spring 2013 www.ntnu.no TMA4115 - Calculus 3, Lecture 21 Review of last week s lecture Last week
More informationMATH 423 Linear Algebra II Lecture 33: Diagonalization of normal operators.
MATH 423 Linear Algebra II Lecture 33: Diagonalization of normal operators. Adjoint operator and adjoint matrix Given a linear operator L on an inner product space V, the adjoint of L is a transformation
More informationMath 504 (Fall 2011) 1. (*) Consider the matrices
Math 504 (Fall 2011) Instructor: Emre Mengi Study Guide for Weeks 11-14 This homework concerns the following topics. Basic definitions and facts about eigenvalues and eigenvectors (Trefethen&Bau, Lecture
More informationAutonomous Systems and Stability
LECTURE 8 Autonomous Systems and Stability An autonomous system is a system of ordinary differential equations of the form 1 1 ( 1 ) 2 2 ( 1 ). ( 1 ) or, in vector notation, x 0 F (x) That is to say, an
More informationData Analysis and Manifold Learning Lecture 2: Properties of Symmetric Matrices and Examples
Data Analysis and Manifold Learning Lecture 2: Properties of Symmetric Matrices and Examples Radu Horaud INRIA Grenoble Rhone-Alpes, France Radu.Horaud@inrialpes.fr http://perception.inrialpes.fr/ Outline
More informationAMS526: Numerical Analysis I (Numerical Linear Algebra for Computational and Data Sciences)
AMS526: Numerical Analysis (Numerical Linear Algebra for Computational and Data Sciences) Lecture 14: Eigenvalue Problems; Eigenvalue Revealing Factorizations Xiangmin Jiao Stony Brook University Xiangmin
More informationAnnounce Statistical Motivation Properties Spectral Theorem Other ODE Theory Spectral Embedding. Eigenproblems I
Eigenproblems I CS 205A: Mathematical Methods for Robotics, Vision, and Graphics Doug James (and Justin Solomon) CS 205A: Mathematical Methods Eigenproblems I 1 / 33 Announcements Homework 1: Due tonight
More informationRemark By definition, an eigenvector must be a nonzero vector, but eigenvalue could be zero.
Sec 6 Eigenvalues and Eigenvectors Definition An eigenvector of an n n matrix A is a nonzero vector x such that A x λ x for some scalar λ A scalar λ is called an eigenvalue of A if there is a nontrivial
More informationLecture II: Rigid-Body Physics
Rigid-Body Motion Previously: Point dimensionless objects moving through a trajectory. Today: Objects with dimensions, moving as one piece. 2 Rigid-Body Kinematics Objects as sets of points. Relative distances
More informationCOMP 558 lecture 18 Nov. 15, 2010
Least squares We have seen several least squares problems thus far, and we will see more in the upcoming lectures. For this reason it is good to have a more general picture of these problems and how to
More informationManipulator Dynamics 2. Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Manipulator Dynamics 2 Forward Dynamics Problem Given: Joint torques and links geometry, mass, inertia, friction Compute: Angular acceleration of the links (solve differential equations) Solution Dynamic
More informationNOTES ON LINEAR ALGEBRA CLASS HANDOUT
NOTES ON LINEAR ALGEBRA CLASS HANDOUT ANTHONY S. MAIDA CONTENTS 1. Introduction 2 2. Basis Vectors 2 3. Linear Transformations 2 3.1. Example: Rotation Transformation 3 4. Matrix Multiplication and Function
More informationPseudospectra and Nonnormal Dynamical Systems
Pseudospectra and Nonnormal Dynamical Systems Mark Embree and Russell Carden Computational and Applied Mathematics Rice University Houston, Texas ELGERSBURG MARCH 1 Overview of the Course These lectures
More informationLecture 2: Linear Algebra Review
EE 227A: Convex Optimization and Applications January 19 Lecture 2: Linear Algebra Review Lecturer: Mert Pilanci Reading assignment: Appendix C of BV. Sections 2-6 of the web textbook 1 2.1 Vectors 2.1.1
More informationEE263: Introduction to Linear Dynamical Systems Review Session 5
EE263: Introduction to Linear Dynamical Systems Review Session 5 Outline eigenvalues and eigenvectors diagonalization matrix exponential EE263 RS5 1 Eigenvalues and eigenvectors we say that λ C is an eigenvalue
More informationAnnounce Statistical Motivation ODE Theory Spectral Embedding Properties Spectral Theorem Other. Eigenproblems I
Eigenproblems I CS 205A: Mathematical Methods for Robotics, Vision, and Graphics Doug James (and Justin Solomon) CS 205A: Mathematical Methods Eigenproblems I 1 / 33 Announcements Homework 1: Due tonight
More informationa 11 a 12 a 11 a 12 a 13 a 21 a 22 a 23 . a 31 a 32 a 33 a 12 a 21 a 23 a 31 a = = = = 12
24 8 Matrices Determinant of 2 2 matrix Given a 2 2 matrix [ ] a a A = 2 a 2 a 22 the real number a a 22 a 2 a 2 is determinant and denoted by det(a) = a a 2 a 2 a 22 Example 8 Find determinant of 2 2
More informationspring, math 204 (mitchell) list of theorems 1 Linear Systems Linear Transformations Matrix Algebra
spring, 2016. math 204 (mitchell) list of theorems 1 Linear Systems THEOREM 1.0.1 (Theorem 1.1). Uniqueness of Reduced Row-Echelon Form THEOREM 1.0.2 (Theorem 1.2). Existence and Uniqueness Theorem THEOREM
More informationThe geometry of least squares
The geometry of least squares We can think of a vector as a point in space, where the elements of the vector are the coordinates of the point. Consider for example, the following vector s: t = ( 4, 0),
More information