# Iterated maps and eigenvalues and vectors

1. Jul 29, 2008

### franky2727

Totaly stuck on this one cant even start to fathom an attempt. first part of the question is show that the eigenvalues of the matrix (2x2 left to right) 4,-1,-4,4 are sigma1=2 and sigma2 =2 and eigenvectors e1= (1,2)T and e2=(1,-2)T done this no problem but am writing this as the second part of the question says

use this result to find the solution to the 2 dimensional linear map

Xn+1 =4Xn -Yn
Yn+1 =-4Xn+4Yn

with X0=1 and y0=1

Last edited: Jul 29, 2008
2. Jul 29, 2008

### Dick

Call v_n the vector (x_n,y_n) and M your given matrix. Then the iteration is v_n+1=M*v_n. So v_n=M^n*v_0. You'll find this easy to express explicitly if you write v_0=(1,1) as a linear combination of eigenvectors of M.