- In this exercise we outline a proof of the PerroniFrobenius Theorem about matrices with positive entries. Let A = (02\") be an N x N matrix with (1,3; > 0 for all 2',j. For vectors 1] = (1:1,... ,uN) and 1') = ('01,... ,1)") we write ti. 2 1') if "a" 2 vi for each i and 11 > 5 if "u.i > \"ui for each i. We write (J: (0,... ,0). (a) Show that if fr 2 0 and i3 a (I, then A17 > fl. 1) For any vector > 6, let 9(17) be the largest A such that AT: 2 A6. (b) Show that 9(6) > 0 for any nonzero I": 2 fl and ifc > 0 then 9(017) = 9(6). Let a = sup 9(17), where the supremum is over all nonzero t") 2 0. By (b) we can consider the supremum over all '0 with llvll = W=1. By continuity of the function g on {\"le = 1} it can be shown that there exists at least one vector 1') 2 0 with 9(6) = a. (c) Show that for any 1') with 9(17) = or, AI": = M), i.e., i": is an eigenvector with eigenvalue 0:. [Hint: we know by denition that At": 2 m"). Assume that they are not equal and consider ALA?) 0:37], using (a).] (d) Show that there is a unique 1': Z (1 with 9(17) 2 a and 2:11)\"; 2 1. [Hint: assume there were two such vectors, "61,172, and consider 9&7] 1'12) and g(|t'11 62|) where |17|=(|'U1|:--- ,Iv\"l)- l (e) Show that all the components of the \"E in (c) are strictly positive. [Hint: if A 2 AI} then A(A) 2 AAJ (f) Show that if A is any other eigenvalue of A, then |A| 0, and hence that a is a simple eigenvalue for A. (j) Explain why every stochastic matrix with strictly positive entries has a unique invariant probability with all positive components. (Apply the above results to the transpose of the stochastic matrix.)