# Up to this point, our main theoretical tools for finding eigenvalues without using det{a λi} = 0 have been the trace and determinant formulas

Save this PDF as:

Size: px
Start display at page:

Download "Up to this point, our main theoretical tools for finding eigenvalues without using det{a λi} = 0 have been the trace and determinant formulas" ## Transcription

1 Finding Eigenvalues Up to this point, our main theoretical tools for finding eigenvalues without using det{a λi} = 0 have been the trace and determinant formulas plus the facts that det{a} = λ λ λ n, Tr{A} = λ +λ + +λ n the eigenvalues of a triangular matrix are the diagonal elements similar matrices B = S AS have the same eigenvalues the eigenvalues of a real symmetric matrix are real the eigenvalues of an orthogonal matrix have λ =. The first fact can be generalized to block triangular matrices. If M is block upper triangular [ A B M = 0 C the eigenvalues of M are the eigenvalues of A plus the eigenvalues of C since [ A λi B 0 = det{m λi} = det = det{a λi}det{c λi}. 0 C λi Here each identity matrix I has the appropriate dimensions to match its partner. A similar statement holds for block lower triangular M. Thus for block triangular matrices, the eigenvalue problem can be broken up into smaller subproblems. (A matrix M is reducible if it can be written in block triangular form by reorderingrowsandcolumnspmp, wherep isapermutationmatrix.) Gershgorin Circle Theorem Also called the Gershgorin Disk Theorem. Theorem statement and Examples and are based on LeVeque s Finite Difference Methods for Ordinary & Partial Differential Equations.

2 Theorem. Let A C n n and let D i be the closed disk in the complex plane centered at A ii with radius given by the row sum r i = j i A ij : D i = {z C : z A ii r i } D(A ii,r = r i ). ThenalltheeigenvaluesofAlieintheunionofthedisksD i fori =,..., n. If some set of k overlapping disks is disjoint from all the other disks, then exactly k eigenvalues lie in the union of these k disks. Note the following: If a disk D i is disjoint from all other disks, then it contains exactly one eigenvalue of A. If a disk D i overlaps other disks, then it need not contain any eigenvalues, although the union of the overlapping disks contains the appropriate number. If A is real, A T has the same eigenvalues as A and the theorem can also be applied to A T (or equivalently the disk radii can be defined by summing the magnitudes of the off-diagonal elements of columns of A rather than rows). (If A is irreducible, a stronger version of the theorem states that an eigenvalue cannot lie on the boundary of a disk unless it lies on the boundary of every disk.) Proof: Ax = λx implies (λ A ii )x i = j i A ij x j λ A ii j i by choosing i such that x i = max j x j. A ij x j x i A ij = r i j i

3 Examples Here we assume all matrices are real example A = The Gershgorin theorem applied to A implies that the eigenvalues lie within the union of D(5,r = 0.7), D(6,r =.), and D(,r = ). There is exactly one eigenvalue in D(,r = ) and two eigenvalues in D(5,r = 0.7) D(6,r =.). All eigenvalues have real parts between and 7. (and hence positive real parts, in particular). The eigenvalue in D(, r = ) must be real, since complex eigenvalues must appear in conjugate pairs. Applying the theorem to A T gives a tighter bound on the single eigenvalue λ near ; λ must lie within D(,r = 0.), so.8 λ.. The actual eigenvalues of A are λ =.9639 and λ ± = 5.580±0.64i.. Second difference matrix ThematrixD () = tridiag[ issymmetric, soallitseigenvaluesarereal. By the Gershgorin theorem they must lie in the circle of radius centered at : 4 λ j 0. We know that D () is nonsingular, so all the eigenvalues of D () are negative. (Also since D () is irreducible and two circles have radii =, 4 < λ j < 0, which proves D () is nonsingular.) The eigenvalues can actually be calculated: λ j = (cos(jπh) ), j =,..., n where h = /(n+), and thus are distributed between 4 < λ j < 0. The Jacobi iteration matrix is B = tridiag[ 0 with < λ j < (since B is irreducible), and Jacobi converges since ρ(b) <. 3. Absolute row sums < Suppose B has absolute row sums <. Then r i = B i + + B i, + B i+, + + B in < B ii 3

4 and by the Gershgorin theorem all λ i < and ρ(b) <. 4. Diagonally dominant matrix Suppose A is diagonally dominant: r i = A i + + A i, + A i+, + + A in < A ii, i =,..., n. Then by the Gershgorin theorem λ i 0 and A is invertible. For Jacobi iteration, B = I D A has all absolute row sums <, so Jacobi converges. Gauss-Seidel and SOR are also guaranteed to converge. In Examples 5 9, see how much information you can extract about the actual eigenvalues using just the Gershgorin Circle Theorem and (block) triangularity. The actual eigenvalues are given. 5. From HW6 λ = 0, λ ± = ± /. 6. From HW7 λ ± = (± 5)/. A = A = [ 0 Last examples are from Theory of Iterative Methods notes 7. For B GS = B J = [ A = [ 0 [ , λ ± = ±, ρ J =, λ = 0, λ = 4, ρ GS = 4 8. Example where Jacobi converges but Gauss-Seidel diverges 0 A =, B J = 0 0 4

5 λ = λ = λ 3 = 0, ρ J = 0, Jacobi converges. 0 B GS = λ = 0, λ = λ 3 =, ρ GS =, Gauss-Seidel diverges. 9. Example where Jacobi diverges but Gauss-Seidel converges A =, B J = λ =, λ = λ 3 = 0.5, ρ J =, Jacobi diverges. B GS = λ = 0, λ ± = 0.35±0.654i, ρ GS = , Gauss-Seidel converges. 5

### Theory of Iterative Methods Based on Strang s Introduction to Applied Mathematics Theory of Iterative Methods The Iterative Idea To solve Ax = b, write Mx (k+1) = (M A)x (k) + b, k = 0, 1,,... Then the error e (k) x (k) x satisfies

### Lecture 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

### c 1995 Society for Industrial and Applied Mathematics Vol. 37, No. 1, pp , March SIAM REVIEW. c 1995 Society for Industrial and Applied Mathematics Vol. 37, No. 1, pp. 93 97, March 1995 008 A UNIFIED PROOF FOR THE CONVERGENCE OF JACOBI AND GAUSS-SEIDEL METHODS * ROBERTO BAGNARA Abstract.

### CHAPTER 5. Basic Iterative Methods Basic Iterative Methods CHAPTER 5 Solve Ax = f where A is large and sparse (and nonsingular. Let A be split as A = M N in which M is nonsingular, and solving systems of the form Mz = r is much easier than

### Scientific Computing WS 2018/2019. Lecture 9. Jürgen Fuhrmann Lecture 9 Slide 1 Scientific Computing WS 2018/2019 Lecture 9 Jürgen Fuhrmann juergen.fuhrmann@wias-berlin.de Lecture 9 Slide 1 Lecture 9 Slide 2 Simple iteration with preconditioning Idea: Aû = b iterative scheme û = û

### Applied Mathematics 205. Unit V: Eigenvalue Problems. Lecturer: Dr. David Knezevic Applied Mathematics 205 Unit V: Eigenvalue Problems Lecturer: Dr. David Knezevic Unit V: Eigenvalue Problems Chapter V.2: Fundamentals 2 / 31 Eigenvalues and Eigenvectors Eigenvalues and eigenvectors of

### Algebra C Numerical Linear Algebra Sample Exam Problems Algebra C Numerical Linear Algebra Sample Exam Problems Notation. Denote by V a finite-dimensional Hilbert space with inner product (, ) and corresponding norm. The abbreviation SPD is used for symmetric

### Bindel, Fall 2016 Matrix Computations (CS 6210) Notes for 1 Iteration basics Notes for 2016-11-07 An iterative solver for Ax = b is produces a sequence of approximations x (k) x. We always stop after finitely many steps, based on some convergence criterion, e.g.

### CAAM 454/554: Stationary Iterative Methods CAAM 454/554: Stationary Iterative Methods Yin Zhang (draft) CAAM, Rice University, Houston, TX 77005 2007, Revised 2010 Abstract Stationary iterative methods for solving systems of linear equations are

### JACOBI S ITERATION METHOD ITERATION METHODS These are methods which compute a sequence of progressively accurate iterates to approximate the solution of Ax = b. We need such methods for solving many large linear systems. Sometimes

### CS 246 Review of Linear Algebra 01/17/19 1 Linear algebra In this section we will discuss vectors and matrices. We denote the (i, j)th entry of a matrix A as A ij, and the ith entry of a vector as v i. 1.1 Vectors and vector operations A vector

### Preliminary/Qualifying Exam in Numerical Analysis (Math 502a) Spring 2012 Instructions Preliminary/Qualifying Exam in Numerical Analysis (Math 502a) Spring 2012 The exam consists of four problems, each having multiple parts. You should attempt to solve all four problems. 1.

### Gershgorin s Circle Theorem for Estimating the Eigenvalues of a Matrix with Known Error Bounds Gershgorin s Circle Theorem for Estimating the Eigenvalues of a Matrix with Known Error Bounds Author: David Marquis Advisors: Professor Hans De Moor Dr. Kathryn Porter Reader: Dr. Michael Nathanson May

### Math 5630: Iterative Methods for Systems of Equations Hung Phan, UMass Lowell March 22, 2018 1 Linear Systems Math 5630: Iterative Methods for Systems of Equations Hung Phan, UMass Lowell March, 018 Consider the system 4x y + z = 7 4x 8y + z = 1 x + y + 5z = 15. We then obtain x = 1 4 (7 + y z)

### Chapter 12: Iterative Methods ES 40: Scientific and Engineering Computation. Uchechukwu Ofoegbu Temple University Chapter : Iterative Methods ES 40: Scientific and Engineering Computation. Gauss-Seidel Method The Gauss-Seidel method

### The convergence of stationary iterations with indefinite splitting The convergence of stationary iterations with indefinite splitting Michael C. Ferris Joint work with: Tom Rutherford and Andy Wathen University of Wisconsin, Madison 6th International Conference on Complementarity

### Solving Linear Systems Solving Linear Systems Iterative Solutions Methods Philippe B. Laval KSU Fall 207 Philippe B. Laval (KSU) Linear Systems Fall 207 / 2 Introduction We continue looking how to solve linear systems of the

### Chapters 5 & 6: Theory Review: Solutions Math 308 F Spring 2015 Chapters 5 & 6: Theory Review: Solutions Math 308 F Spring 205. If A is a 3 3 triangular matrix, explain why det(a) is equal to the product of entries on the diagonal. If A is a lower triangular or diagonal

### Recall : 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

### 9. Iterative Methods for Large Linear Systems EE507 - Computational Techniques for EE Jitkomut Songsiri 9. Iterative Methods for Large Linear Systems introduction splitting method Jacobi method Gauss-Seidel method successive overrelaxation (SOR) 9-1

### Numerical Methods - Numerical Linear Algebra Numerical Methods - Numerical Linear Algebra Y. K. Goh Universiti Tunku Abdul Rahman 2013 Y. K. Goh (UTAR) Numerical Methods - Numerical Linear Algebra I 2013 1 / 62 Outline 1 Motivation 2 Solving Linear

### Computational Linear Algebra Computational Linear Algebra PD Dr. rer. nat. habil. Ralf Peter Mundani Computation in Engineering / BGU Scientific Computing in Computer Science / INF Winter Term 2017/18 Part 3: Iterative Methods PD

### Solving Linear Systems of Equations November 6, 2013 Introduction The type of problems that we have to solve are: Solve the system: A x = B, where a 11 a 1N a 12 a 2N A =.. a 1N a NN x = x 1 x 2. x N B = b 1 b 2. b N To find A 1 (inverse

### Lecture Note 7: Iterative methods for solving linear systems. Xiaoqun Zhang Shanghai Jiao Tong University Lecture Note 7: Iterative methods for solving linear systems Xiaoqun Zhang Shanghai Jiao Tong University Last updated: December 24, 2014 1.1 Review on linear algebra Norms of vectors and matrices vector

### Lecture 18 Classical Iterative Methods Lecture 18 Classical Iterative Methods MIT 18.335J / 6.337J Introduction to Numerical Methods Per-Olof Persson November 14, 2006 1 Iterative Methods for Linear Systems Direct methods for solving Ax = b,

### Iterative Methods. Splitting Methods Iterative Methods Splitting Methods 1 Direct Methods Solving Ax = b using direct methods. Gaussian elimination (using LU decomposition) Variants of LU, including Crout and Doolittle Other decomposition

### Numerical Linear Algebra Homework Assignment - Week 2 Numerical Linear Algebra Homework Assignment - Week 2 Đoàn Trần Nguyên Tùng Student ID: 1411352 8th October 2016 Exercise 2.1: Show that if a matrix A is both triangular and unitary, then it is diagonal.

### Department of Mathematics California State University, Los Angeles Master s Degree Comprehensive Examination in. NUMERICAL ANALYSIS Spring 2015 Department of Mathematics California State University, Los Angeles Master s Degree Comprehensive Examination in NUMERICAL ANALYSIS Spring 2015 Instructions: Do exactly two problems from Part A AND two

### Question: Given an n x n matrix A, how do we find its eigenvalues? Idea: Suppose c is an eigenvalue of A, then what is the determinant of A-cI? Section 5. The Characteristic Polynomial Question: Given an n x n matrix A, how do we find its eigenvalues? Idea: Suppose c is an eigenvalue of A, then what is the determinant of A-cI? Property The eigenvalues

### Math 471 (Numerical methods) Chapter 3 (second half). System of equations Math 47 (Numerical methods) Chapter 3 (second half). System of equations Overlap 3.5 3.8 of Bradie 3.5 LU factorization w/o pivoting. Motivation: ( ) A I Gaussian Elimination (U L ) where U is upper triangular

### Math 240 Calculus III The Calculus III Summer 2015, Session II Wednesday, July 8, 2015 Agenda 1. of the determinant 2. determinants 3. of determinants What is the determinant? Yesterday: Ax = b has a unique solution when A

### Jordan Journal of Mathematics and Statistics (JJMS) 5(3), 2012, pp A NEW ITERATIVE METHOD FOR SOLVING LINEAR SYSTEMS OF EQUATIONS Jordan Journal of Mathematics and Statistics JJMS) 53), 2012, pp.169-184 A NEW ITERATIVE METHOD FOR SOLVING LINEAR SYSTEMS OF EQUATIONS ADEL H. AL-RABTAH Abstract. The Jacobi and Gauss-Seidel iterative

### Chapter 3. Determinants and Eigenvalues Chapter 3. Determinants and Eigenvalues 3.1. Determinants With each square matrix we can associate a real number called the determinant of the matrix. Determinants have important applications to the theory

### Here is an example of a block diagonal matrix with Jordan Blocks on the diagonal: J Class Notes 4: THE SPECTRAL RADIUS, NORM CONVERGENCE AND SOR. Math 639d Due Date: Feb. 7 (updated: February 5, 2018) In the first part of this week s reading, we will prove Theorem 2 of the previous class.

### Warm-up. True or false? Baby proof. 2. The system of normal equations for A x = y has solutions iff A x = y has solutions Warm-up True or false? 1. proj u proj v u = u 2. The system of normal equations for A x = y has solutions iff A x = y has solutions 3. The normal equations are always consistent Baby proof 1. Let A be

### MAT 1332: CALCULUS FOR LIFE SCIENCES. Contents. 1. Review: Linear Algebra II Vectors and matrices Definition. 1.2. MAT 1332: CALCULUS FOR LIFE SCIENCES JING LI Contents 1 Review: Linear Algebra II Vectors and matrices 1 11 Definition 1 12 Operations 1 2 Linear Algebra III Inverses and Determinants 1 21 Inverse Matrices

### 6. Iterative Methods for Linear Systems. The stepwise approach to the solution... 6 Iterative Methods for Linear Systems The stepwise approach to the solution Miriam Mehl: 6 Iterative Methods for Linear Systems The stepwise approach to the solution, January 18, 2013 1 61 Large Sparse

### Eigenvalues and Eigenvectors Chapter 1 Eigenvalues and Eigenvectors Among problems in numerical linear algebra, the determination of the eigenvalues and eigenvectors of matrices is second in importance only to the solution of linear

### Computing Eigenvalues and/or Eigenvectors;Part 1, Generalities and symmetric matrices Computing Eigenvalues and/or Eigenvectors;Part 1, Generalities and symmetric matrices Tom Lyche Centre of Mathematics for Applications, Department of Informatics, University of Oslo November 9, 2008 Today

### Introduction to Scientific Computing (Lecture 5: Linear system of equations / Matrix Splitting) Bojana Rosić, Thilo Moshagen Institute of Scientific Computing Motivation Let us resolve the problem scheme by using Kirchhoff s laws: the algebraic

### Math/Phys/Engr 428, Math 529/Phys 528 Numerical Methods - Summer Homework 3 Due: Tuesday, July 3, 2018 Math/Phys/Engr 428, Math 529/Phys 528 Numerical Methods - Summer 28. (Vector and Matrix Norms) Homework 3 Due: Tuesday, July 3, 28 Show that the l vector norm satisfies the three properties (a) x for x

### 9.1 Preconditioned Krylov Subspace Methods Chapter 9 PRECONDITIONING 9.1 Preconditioned Krylov Subspace Methods 9.2 Preconditioned Conjugate Gradient 9.3 Preconditioned Generalized Minimal Residual 9.4 Relaxation Method Preconditioners 9.5 Incomplete

### Matrix Operations: Determinant Matrix Operations: Determinant Determinants Determinants are only applicable for square matrices. Determinant of the square matrix A is denoted as: det(a) or A Recall that the absolute value of the determinant

### Numerical Analysis: Solutions of System of. Linear Equation. Natasha S. Sharma, PhD Mathematical Question we are interested in answering numerically How to solve the following linear system for x Ax = b? where A is an n n invertible matrix and b is vector of length n. Notation: x denote

### Numerical Programming I (for CSE) Technische Universität München WT 1/13 Fakultät für Mathematik Prof. Dr. M. Mehl B. Gatzhammer January 1, 13 Numerical Programming I (for CSE) Tutorial 1: Iterative Methods 1) Relaxation Methods a) Let

### Foundations of Matrix Analysis 1 Foundations of Matrix Analysis In this chapter we recall the basic elements of linear algebra which will be employed in the remainder of the text For most of the proofs as well as for the details, the

### 1. Linear systems of equations. Chapters 7-8: Linear Algebra. Solution(s) of a linear system of equations (continued) 1 A linear system of equations of the form Sections 75, 78 & 81 a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2 a m1 x 1 + a m2 x 2 + + a mn x n = b m can be written in matrix

### Dimension. Eigenvalue and eigenvector Dimension. Eigenvalue and eigenvector Math 112, week 9 Goals: Bases, dimension, rank-nullity theorem. Eigenvalue and eigenvector. Suggested Textbook Readings: Sections 4.5, 4.6, 5.1, 5.2 Week 9: Dimension,

### 10.2 ITERATIVE METHODS FOR SOLVING LINEAR SYSTEMS. The Jacobi Method 54 CHAPTER 10 NUMERICAL METHODS 10. ITERATIVE METHODS FOR SOLVING LINEAR SYSTEMS As a numerical technique, Gaussian elimination is rather unusual because it is direct. That is, a solution is obtained after

### Computing Eigenvalues and/or Eigenvectors;Part 1, Generalities and symmetric matrices Computing Eigenvalues and/or Eigenvectors;Part 1, Generalities and symmetric matrices Tom Lyche Centre of Mathematics for Applications, Department of Informatics, University of Oslo November 8, 2009 Today

### Math 108b: Notes on the Spectral Theorem Math 108b: Notes on the Spectral Theorem From section 6.3, we know that every linear operator T on a finite dimensional inner product space V has an adjoint. (T is defined as the unique linear operator

### Eigenvalue and Eigenvector Problems Eigenvalue and Eigenvector Problems An attempt to introduce eigenproblems Radu Trîmbiţaş Babeş-Bolyai University April 8, 2009 Radu Trîmbiţaş ( Babeş-Bolyai University) Eigenvalue and Eigenvector Problems

### Econ 204 Supplement to Section 3.6 Diagonalization and Quadratic Forms. 1 Diagonalization and Change of Basis Econ 204 Supplement to Section 3.6 Diagonalization and Quadratic Forms De La Fuente notes that, if an n n matrix has n distinct eigenvalues, it can be diagonalized. In this supplement, we will provide

### Chapter 5 Eigenvalues and Eigenvectors Chapter 5 Eigenvalues and Eigenvectors Outline 5.1 Eigenvalues and Eigenvectors 5.2 Diagonalization 5.3 Complex Vector Spaces 2 5.1 Eigenvalues and Eigenvectors Eigenvalue and Eigenvector If A is a n n

### Lecture 4 Eigenvalue problems Lecture 4 Eigenvalue problems Weinan E 1,2 and Tiejun Li 2 1 Department of Mathematics, Princeton University, weinan@princeton.edu 2 School of Mathematical Sciences, Peking University, tieli@pku.edu.cn

### Eigenvalues and Eigenvectors 5 Eigenvalues and Eigenvectors 5.2 THE CHARACTERISTIC EQUATION DETERMINANATS nn Let A be an matrix, let U be any echelon form obtained from A by row replacements and row interchanges (without scaling),

### EE5120 Linear Algebra: Tutorial 6, July-Dec Covers sec 4.2, 5.1, 5.2 of GS EE0 Linear Algebra: Tutorial 6, July-Dec 07-8 Covers sec 4.,.,. of GS. State True or False with proper explanation: (a) All vectors are eigenvectors of the Identity matrix. (b) Any matrix can be diagonalized.

### Goal: to construct some general-purpose algorithms for solving systems of linear Equations Chapter IV Solving Systems of Linear Equations Goal: to construct some general-purpose algorithms for solving systems of linear Equations 4.6 Solution of Equations by Iterative Methods 4.6 Solution of

### Math 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:

### Chapter 7 Iterative Techniques in Matrix Algebra Chapter 7 Iterative Techniques in Matrix Algebra Per-Olof Persson persson@berkeley.edu Department of Mathematics University of California, Berkeley Math 128B Numerical Analysis Vector Norms Definition

### A NEW EFFECTIVE PRECONDITIONED METHOD FOR L-MATRICES Journal of Mathematical Sciences: Advances and Applications Volume, Number 2, 2008, Pages 3-322 A NEW EFFECTIVE PRECONDITIONED METHOD FOR L-MATRICES Department of Mathematics Taiyuan Normal University

### Example: Filter output power. maximization. Definition. Eigenvalues, eigenvectors and similarity. Example: Stability of linear systems. Lecture 2: Eigenvalues, eigenvectors and similarity The single most important concept in matrix theory. German word eigen means proper or characteristic. KTH Signal Processing 1 Magnus Jansson/Emil Björnson

### Econ Slides from Lecture 7 Econ 205 Sobel Econ 205 - Slides from Lecture 7 Joel Sobel August 31, 2010 Linear Algebra: Main Theory A linear combination of a collection of vectors {x 1,..., x k } is a vector of the form k λ ix i for

### Determinants by Cofactor Expansion (III) Determinants by Cofactor Expansion (III) Comment: (Reminder) If A is an n n matrix, then the determinant of A can be computed as a cofactor expansion along the jth column det(a) = a1j C1j + a2j C2j +...

### vibrations, light transmission, tuning guitar, design buildings and bridges, washing machine, Partial differential problems, water flow,... 6 Eigenvalues Eigenvalues are a common part of our life: vibrations, light transmission, tuning guitar, design buildings and bridges, washing machine, Partial differential problems, water flow, The simplest

### Chap 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

### EIGENVALUE PROBLEMS. Background on eigenvalues/ eigenvectors / decompositions. Perturbation analysis, condition numbers.. EIGENVALUE PROBLEMS Background on eigenvalues/ eigenvectors / decompositions Perturbation analysis, condition numbers.. Power method The QR algorithm Practical QR algorithms: use of Hessenberg form and

### Solving Linear Systems Solving Linear Systems Iterative Solutions Methods Philippe B. Laval KSU Fall 2015 Philippe B. Laval (KSU) Linear Systems Fall 2015 1 / 12 Introduction We continue looking how to solve linear systems of

### c c c c c c c c c c a 3x3 matrix C= has a determinant determined by Linear Algebra Determinants and Eigenvalues Introduction: Many important geometric and algebraic properties of square matrices are associated with a single real number revealed by what s known as the determinant.

### Chapter 2. Square matrices Chapter 2. Square matrices Lecture notes for MA1111 P. Karageorgis pete@maths.tcd.ie 1/18 Invertible matrices Definition 2.1 Invertible matrices An n n matrix A is said to be invertible, if there is a

### The Eigenvalue Problem: Perturbation Theory Jim Lambers MAT 610 Summer Session 2009-10 Lecture 13 Notes These notes correspond to Sections 7.2 and 8.1 in the text. The Eigenvalue Problem: Perturbation Theory The Unsymmetric Eigenvalue Problem Just

### and let s calculate the image of some vectors under the transformation T. Chapter 5 Eigenvalues and Eigenvectors 5. Eigenvalues and Eigenvectors Let T : R n R n be a linear transformation. Then T can be represented by a matrix (the standard matrix), and we can write T ( v) =

### Math Introduction to Numerical Analysis - Class Notes. Fernando Guevara Vasquez. Version Date: January 17, 2012. Math 5620 - Introduction to Numerical Analysis - Class Notes Fernando Guevara Vasquez Version 1990. Date: January 17, 2012. 3 Contents 1. Disclaimer 4 Chapter 1. Iterative methods for solving linear systems

### CLASSICAL ITERATIVE METHODS CLASSICAL ITERATIVE METHODS LONG CHEN In this notes we discuss classic iterative methods on solving the linear operator equation (1) Au = f, posed on a finite dimensional Hilbert space V = R N equipped

### Dominant Eigenvalue of a Sudoku Submatrix Sacred Heart University DigitalCommons@SHU Academic Festival Apr 20th, 9:30 AM - 10:45 AM Dominant Eigenvalue of a Sudoku Submatrix Nicole Esposito Follow this and additional works at: https://digitalcommons.sacredheart.edu/acadfest

### Conjugate 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

### 9.1 Eigenvectors and Eigenvalues of a Linear Map Chapter 9 Eigenvectors and Eigenvalues 9.1 Eigenvectors and Eigenvalues of a Linear Map Given a finite-dimensional vector space E, letf : E! E be any linear map. If, by luck, there is a basis (e 1,...,e

### Chapter 5. Eigenvalues and Eigenvectors Chapter 5 Eigenvalues and Eigenvectors Section 5. Eigenvectors and Eigenvalues Motivation: Difference equations A Biology Question How to predict a population of rabbits with given dynamics:. half of the

### A LINEAR SYSTEMS OF EQUATIONS. By : Dewi Rachmatin A LINEAR SYSTEMS OF EQUATIONS By : Dewi Rachmatin Back Substitution We will now develop the backsubstitution algorithm, which is useful for solving a linear system of equations that has an upper-triangular

### AMS526: Numerical Analysis I (Numerical Linear Algebra) AMS526: Numerical Analysis I (Numerical Linear Algebra) Lecture 21: Sensitivity of Eigenvalues and Eigenvectors; Conjugate Gradient Method Xiangmin Jiao Stony Brook University Xiangmin Jiao Numerical Analysis

### Computational Methods. Systems of Linear Equations Computational Methods Systems of Linear Equations Manfred Huber 2010 1 Systems of Equations Often a system model contains multiple variables (parameters) and contains multiple equations Multiple equations

### Classical iterative methods for linear systems Classical iterative methods for linear systems Ed Bueler MATH 615 Numerical Analysis of Differential Equations 27 February 1 March, 2017 Ed Bueler (MATH 615 NADEs) Classical iterative methods for linear

### SyDe312 (Winter 2005) Unit 1 - Solutions (continued) SyDe3 (Winter 5) Unit - Solutions (continued) March, 5 Chapter 6 - Linear Systems Problem 6.6 - b Iterative solution by the Jacobi and Gauss-Seidel iteration methods: Given: b = [ 77] T, x = [ ] T 9x +

### AMS526: Numerical Analysis I (Numerical Linear Algebra) AMS526: Numerical Analysis I (Numerical Linear Algebra) Lecture 16: Eigenvalue Problems; Similarity Transformations Xiangmin Jiao Stony Brook University Xiangmin Jiao Numerical Analysis I 1 / 18 Eigenvalue

### Notes on Linear Algebra and Matrix Theory Massimo Franceschet featuring Enrico Bozzo Scalar product The scalar product (a.k.a. dot product or inner product) of two real vectors x = (x 1,..., x n ) and y = (y 1,..., y n ) is not a vector but a

### Linear 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

### Matrices and Linear Algebra Contents Quantitative methods for Economics and Business University of Ferrara Academic year 2017-2018 Contents 1 Basics 2 3 4 5 Contents 1 Basics 2 3 4 5 Contents 1 Basics 2 3 4 5 Contents 1 Basics 2

### Today s class. Linear Algebraic Equations LU Decomposition. Numerical Methods, Fall 2011 Lecture 8. Prof. Jinbo Bi CSE, UConn Today s class Linear Algebraic Equations LU Decomposition 1 Linear Algebraic Equations Gaussian Elimination works well for solving linear systems of the form: AX = B What if you have to solve the linear

### MATH 1553-C MIDTERM EXAMINATION 3 MATH 553-C MIDTERM EXAMINATION 3 Name GT Email @gatech.edu Please read all instructions carefully before beginning. Please leave your GT ID card on your desk until your TA scans your exam. Each problem

### MAT 610: Numerical Linear Algebra. James V. Lambers MAT 610: Numerical Linear Algebra James V Lambers January 16, 2017 2 Contents 1 Matrix Multiplication Problems 7 11 Introduction 7 111 Systems of Linear Equations 7 112 The Eigenvalue Problem 8 12 Basic

### Name: MATH 3195 :: Fall 2011 :: Exam 2. No document, no calculator, 1h00. Explanations and justifications are expected for full credit. Name: MATH 3195 :: Fall 2011 :: Exam 2 No document, no calculator, 1h00. Explanations and justifications are expected for full credit. 1. ( 4 pts) Say which matrix is in row echelon form and which is not.

### LINEAR ALGEBRA SUMMARY SHEET. LINEAR ALGEBRA SUMMARY SHEET RADON ROSBOROUGH https://intuitiveexplanationscom/linear-algebra-summary-sheet/ This document is a concise collection of many of the important theorems of linear algebra, organized

### Math 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 AMSC/CMSC 661 Scientific Computing II Spring 2005 Solution of Sparse Linear Systems Part 2: Iterative methods Dianne P. O Leary c 2005 Solving Sparse Linear Systems: Iterative methods The plan: Iterative ELE/MCE 503 Linear Algebra Facts Fall 2018 Fact N.1 A set of vectors is linearly independent if and only if none of the vectors in the set can be written as a linear combination of the others. Fact N.2