a Prove that x (k) x T k x (0) x and x (k) x T k 1 T x (1) x (0) , Where T is an n n matrix with T 1 and x (k) Tx (k1) c, k 1, 2, , With x (0) arbitrary, c R n , and x Tx c b Apply the bounds to Exercise 1, when possible, using the l norm ...

a Subtract x Txc from x Tx1c to obtain xk x T x1x Thus x x Tx x Inductively ...

a. Prove that ||x (k) x|| ||T|| k ||x (0) x|| and ||x (k)

Question:

a. Prove that ||x^(k) − x|| ≤ ||T||^k ||x⁽⁰⁾ − x|| and ||x^(k) − x|| ≤ ||T||^k /1 − ||T|| ||x⁽¹⁾ − x⁽⁰⁾ ||, Where T is an n × n matrix with ||T|| < 1 and x^(k) = Tx^(k−1) + c, k = 1, 2, . . . , With x⁽⁰⁾ arbitrary, c ∈ Rⁿ, and x = Tx + c.

b. Apply the bounds to Exercise 1, when possible, using the l∞ norm.