Iterated maps and eigenvalues and vectors

[franky2727]

totally stuck on this one can't 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

please help as i have no idea where to even begin. thanks

 

[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.