Suppose that A SAS 1 where is a diagonal matrix with diagonal elements 1, 2, , n (a) Show that ASi iSi i 1, , n (b) Show that if x a1S1 a2s2 ns2 then Akx 1k1s1 2k2s2 nknsn (c) Suppose that i 1 for I 1, n What happens to Akx as k Explain ...

a If A SAS1 then AS AS If s is the ith column of S then As is the ith column of AS nd is is th ...

Suppose that A = SAS-1 where is a diagonal matrix with diagonal elements 1, 2, .

Question:

Suppose that A = SAS-1 where Λ is a diagonal matrix with diagonal elements λ1, λ2, . . . , λn.
(a) Show that ASi = λiSi i = 1,..., n.
(b) Show that if x = a1S1 + a2s2 + αns2 +.....+ then
Akx = α1λk1s1 + α2λk2s2 +......+ αnλknsn
(c) Suppose that |λi| < 1 for I = 1,....n. What happens to Akx as k → ∞? Explain.