MATH 374 Lecture 23 Complex Eigenvalues.

Slides:



Advertisements
Similar presentations
Ch 7.6: Complex Eigenvalues
Advertisements

Ch 7.7: Fundamental Matrices
Chapter 28 – Part II Matrix Operations. Gaussian elimination Gaussian elimination LU factorization LU factorization Gaussian elimination with partial.
Chapter 6 Eigenvalues and Eigenvectors
OCE301 Part II: Linear Algebra lecture 4. Eigenvalue Problem Ax = y Ax = x occur frequently in engineering analysis (eigenvalue problem) Ax =  x [ A.
Some useful linear algebra. Linearly independent vectors span(V): span of vector space V is all linear combinations of vectors v i, i.e.
Ch 7.8: Repeated Eigenvalues
example: four masses on springs
Ch 7.3: Systems of Linear Equations, Linear Independence, Eigenvalues
5 5.1 © 2012 Pearson Education, Inc. Eigenvalues and Eigenvectors EIGENVECTORS AND EIGENVALUES.
1 MAC 2103 Module 12 Eigenvalues and Eigenvectors.
A matrix equation has the same solution set as the vector equation which has the same solution set as the linear system whose augmented matrix is Therefore:
Matrices CHAPTER 8.1 ~ 8.8. Ch _2 Contents  8.1 Matrix Algebra 8.1 Matrix Algebra  8.2 Systems of Linear Algebra Equations 8.2 Systems of Linear.
Linear algebra: matrix Eigen-value Problems
Chapter 5 Eigenvalues and Eigenvectors 大葉大學 資訊工程系 黃鈴玲 Linear Algebra.
Section 5.1 First-Order Systems & Applications
Homogeneous Linear Systems with Constant Coefficients Solutions of Systems of ODEs.
Boyce/DiPrima 9 th ed, Ch 7.6: Complex Eigenvalues Elementary Differential Equations and Boundary Value Problems, 9 th edition, by William E. Boyce and.
Nonhomogeneous Linear Systems Undetermined Coefficients.
5.1 Eigenvectors and Eigenvalues 5. Eigenvalues and Eigenvectors.
1.7 Linear Independence. in R n is said to be linearly independent if has only the trivial solution. in R n is said to be linearly dependent if there.
Linear Algebra Chapter 6 Linear Algebra with Applications -Gareth Williams Br. Joel Baumeyer, F.S.C.
5 5.1 © 2016 Pearson Education, Ltd. Eigenvalues and Eigenvectors EIGENVECTORS AND EIGENVALUES.
Ch 4.2: Homogeneous Equations with Constant Coefficients Consider the nth order linear homogeneous differential equation with constant, real coefficients:
Differential Equations MTH 242 Lecture # 28 Dr. Manshoor Ahmed.
Complex Eigenvalues and Phase Portraits. Fundamental Set of Solutions For Linear System of ODEs With Eigenvalues and Eigenvectors and The General Solution.
Math 4B Systems of Differential Equations Matrix Solutions Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB.
Review of Eigenvectors and Eigenvalues from CliffsNotes Online mining-the-Eigenvectors-of-a- Matrix.topicArticleId-20807,articleId-
Characteristic Polynomial Hung-yi Lee. Outline Last lecture: Given eigenvalues, we know how to find eigenvectors or eigenspaces Check eigenvalues This.
Chapter 6 Eigenvalues and Eigenvectors
Systems of Linear Differential Equations
Review of Eigenvectors and Eigenvalues
Linear Equations Constant Coefficients
Continuum Mechanics (MTH487)
ISHIK UNIVERSITY FACULTY OF EDUCATION Mathematics Education Department
Eigen Decomposition Based on the slides by Mani Thomas and book by Gilbert Strang. Modified and extended by Longin Jan Latecki.
Complex Eigenvalues kshum ENGG2420B.
Solutions to Systems Of Linear Equations II
Boyce/DiPrima 10th ed, Ch 7.7: Fundamental Matrices Elementary Differential Equations and Boundary Value Problems, 10th edition, by William E. Boyce and.
Boyce/DiPrima 10th ed, Ch 7.8: Repeated Eigenvalues Elementary Differential Equations and Boundary Value Problems, 10th edition, by William E. Boyce.
Linear Algebra Lecture 36.
Chapter 5: Linear Equations with Constant Coefficients
Systems of First Order Linear Equations
Eigenvalues and Eigenvectors
Some useful linear algebra
Eigen Decomposition Based on the slides by Mani Thomas and book by Gilbert Strang. Modified and extended by Longin Jan Latecki.
Numerical Analysis Lecture 16.
Eigen Decomposition Based on the slides by Mani Thomas and book by Gilbert Strang. Modified and extended by Longin Jan Latecki.
Ch 4.2: Homogeneous Equations with Constant Coefficients
Boyce/DiPrima 10th ed, Ch 7.3: Systems of Linear Equations, Linear Independence, Eigenvalues Elementary Differential Equations and Boundary Value Problems,
General Solution – Homogeneous and Non-Homogeneous Equations
Linear Algebra Lecture 3.
MATH 374 Lecture 24 Repeated Eigenvalues.
Eigen Decomposition Based on the slides by Mani Thomas and book by Gilbert Strang. Modified and extended by Longin Jan Latecki.
MAE 82 – Engineering Mathematics
Non-Homogeneous Systems
Eigenvalues and Eigenvectors
Linear Algebra Lecture 32.
Eigen Decomposition Based on the slides by Mani Thomas and book by Gilbert Strang. Modified and extended by Longin Jan Latecki.
Boyce/DiPrima 10th ed, Ch 7.6: Complex Eigenvalues Elementary Differential Equations and Boundary Value Problems, 10th edition, by William E. Boyce and.
Linear Algebra Lecture 20.
Elementary Linear Algebra Anton & Rorres, 9th Edition
Linear Algebra Lecture 7.
Homogeneous Linear Systems
Linear Algebra Lecture 30.
Eigenvalues and Eigenvectors
Eigenvalues and Eigenvectors
Linear Algebra: Matrix Eigenvalue Problems – Part 2
Auxiliary Equation with Complex Roots; Hyperbolic Functions
Linear Algebra Lecture 28.
Presentation transcript:

MATH 374 Lecture 23 Complex Eigenvalues

8.5: Complex Eigenvalues Definition: For any complex number z = a+ib, with a and b real numbers, Re(z) = a, Im(z) = b, and the conjugate of z is the complex number Similar definitions hold for complex matrices. For any matrix A = [aij], Ā is the conjugate matrix [āij].

Homogeneous Systems Revisited Consider again the system of n linear homogeneous equations in n unknowns: X’ = AX. (1) Looking for non-trivial solutions of (1) of the form X = Cemt, we find that m must be an eigenvalue of A with corresponding eigenvector C. Recall that m is a root of the characteristic equation |A – mI| = 0 (2) and C is a non-zero solution of (A – mI)C = 0. (3)

Complex Eigenvalue Case Suppose for  and  real numbers, m =  + i is an eigenvalue of matrix A with corresponding eigenvector C.

Complex Eigenvalue Case X’ = AX. (1) (A – mI)C = 0. (3) Complex Eigenvalue Case

Complex Eigenvalue Case

Complex Eigenvalue Case

Complex Eigenvalue Case

Complex Eigenvalue Case

Complex Eigenvalue Case X’ = AX (1) Complex Eigenvalue Case Theorem 8.8: Let m =  + i  be an eigenvalue of the constant real-valued matrix A in (1) with corresponding eigenvector C. Then X1(t) = [Re(C) cost – Im(C) sint]et and X2(t) = [Im(C) cost + Re(C) sint]et are linearly independent solutions of (1).

Example 1

Example 1

Example 1

Example 1