This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: M346 Second Midterm Exam, October 23, 2009 1) Diagonalization: a)(10 pts) Find the eigenvalues and eigenvectors of A = parenleftbigg 3 8 2 3 parenrightbigg b) (10 pts) Compute e Bt , where B = parenleftbigg 1 2 2 4 parenrightbigg . c) (10 pts) Find the eigenvalues of C = 2 2 3 7 2 5 2 8 6 16 4 6 . You do not need to find the eigenvectors. 2. Consider the system of equations x ( n +1) = A x ( n ), where A = parenleftbigg 1 2 1 parenrightbigg , with initial condition x (0) = parenleftbigg 1 5 parenrightbigg . a) (15 pts) Find x ( n ) for all n . Be as explicit as possible. b) (5 pts) Find lim n x 1 ( n ) x 2 ( n ) . c) (5 pts) Find lim n x 1 ( n +1) x 1 ( n ) . 3. a) (15 pts) Consider the system of nonlinear coupled differential equations dx 1 dt = x 1 (3 2 x 1 x 2 ) dx 2 dt = x 2 (3 x 1 2 x 2 ) . This system of equations has four fixed points, namely (0 , 0) T , (3 / 2 , 0) T , (0 , 3 / 2) T , and (1 , 1) T . These equations describe competition between two....
View
Full
Document
This note was uploaded on 04/10/2011 for the course M 346 taught by Professor Radin during the Spring '08 term at University of Texas at Austin.
 Spring '08
 RAdin
 Linear Algebra, Algebra, Eigenvectors, Vectors

Click to edit the document details