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Unformatted text preview: Linear Algebra(Bretscher) Chapter 7 Notes 1 Dynamical Systems and Eigenvectors: An Introductory Exam- ple 1.1 Eigenvectors and Eigenvalues Definition Consider an n × n matrix A . A nonzero vector ~v in R n is called an eigenvector of A if A~v is a scalar multiple of vecv , that is, if A~v = λ~v for some scalar λ . This scalar may be zero. The scalar λ is called the eigenvalue associated with the eigenvector ~v . Theorem 1.1. The possible real eigenvalues of an orthogonal matrix are 1 and- 1 . 1.2 Dynamical Systems and Eigenvectors Theorem 1.2 (Discrete dynamical system) . Consider the dynamical system ~x ( t + 1) = A~x ( t ) with~x (0) = ~x Then ~x ( t ) = A t ~x suppose that we can find a basis ~v 1 ,~v 2 ,...,~v n of R n consisting of eigenvectors of A , with A~v n = λ n ~v n Find the coordinates c 1 ,c 2 ,...,c n of vector ~x with respect to basis ~v 1 ,~v 2 ,...,~v n : ~x = c 1 ~v 1 + c 2 ~v 2 + ... + c n ~v n Then ~x ( t ) = c 1 λ t 1 ~v 1 + c 2 λ t 2 ~v 2 + ... + c n λ t n ~v n Definition Consider a discrete dynamic system ~x ( t + 1) = A~x ( t ) initial value ~x (0) = ~x , where A is a 2 × 2 matrix. In this case, the state vector ~x ( t ) = x 1 ( t ) x 2 ( t ) can be represented geometrically in the x 1- x 2-plane. The endpoints of state vectors vecx (0) = ~x ,vecx (1) = A~x ,... form the discrete trajectory of this system, representing its evolution in the future. A discrete phase portrait of the system ~x ( t + 1) = A~x ( t ) shows trajectories for various initial states, capturing all the qualitatively different scenarios....
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This note was uploaded on 04/27/2010 for the course PSYC 3023 taught by Professor Symons during the Spring '08 term at Acadia.
- Spring '08