Math 2602  A1 - A3                                                                               Homework #11
                                               Due Tuesday, November 22, 2005


         Let (a_n) and (b_n) be the sequences defined by a₀ = 0 , b₀ = 1, and ( a_n , b_n )^T = A( a_n-1 , b_n-1 ) , where

                                       1       2/3
                         A =       1/3    2/3    .

   
        a) Find the eigenvalues and eigenvectors of A.


        b)  Express ( 0 ,1 )^T as a linear combination of eigenvectors of A.


        c)  Use (b) to find a_n and b_n .


       d)  Repeat (b) and (c) for a₀ = 2 and b₀ = 1 .