Let A be the 4x4 matrix with elements in positions (1,2), (2,3), (3,4), and (4,1) equal to 1, and all other elements equal to zero. Let b = e_1 and x_0=0. Show that ||r_k||_2=||r_0||_2 for k=1,2,3.

Here, r_k=Ax_k-b where x_k is the iterate produced by GMRES in iteration k.

Finding r_0 is not a problem for me since x_0 is the zero vector. Therefore, ||r_0||= b. However, we have not learned how to actually figure out the different iterations, so I'm guessing that we do not have to find x_1, x_2, and x_3, explicitly, but I have no idea how to show that ||r_k||_2=b for k=1,2,3. Can anyone push me in the right direction? Thank you.