Hello! I just don't know how to solve i j k, so you may only share your ideas about these three parts. Thank you!

1.20 In this exercise we outline a proof of the Perron-Frobenius Theorem about matrices with positive entries. Let A = (aj ) be an N x N matrix with dij > 0 for all i, j. For vectors u = (u], ..., uN) and v = (v], ... , UN) we write u > v if u' > v' for each i and u > v if u' > v' for each i. We write 0 = (0, . . . ,0). (a) Show that if v 2 0 and v # 0, then Av > 0. For any vector v 2 0, let g(v) be the largest A such that Av 2 1v. (b) Show that g(v) > 0 for any nonzero v 2 0 and if c > 0 then g(cu) = g(v). Let ' (a ) 6 dns = 0 where the supremum is over all nonzero v 2 0. By (b) we can consider the supremum over all v with 11011 = (01)2+ ... + (UN)2 = 1. By continuity of the function g on {lv|| = 1} it can be shown that there exists at least one vector v 2 0 with g(v) = a. (c) Show that for any u with g(v) = a, Av = Qu, i.e., v is an eigenvector with eigenvalue o. [Hint: we know by definition that Av 2 Qv. Assume that they are not equal and consider A[Av - av],using (a) ] (d) Show that there is a unique 1': 2 (j with 9(17) = a and 2,111.? = 1. [Hint: assume there were two such vectors, 51,152, and consider 9W1 \"52) and g(|1 'Eg|) where |5|=(|v1|,... alvnll l (e) Show that all the components of the in (c) are strictly positive. [Hint: if A17 2 Air then A(A) 2 AAR] (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 r$ults to the transpose of the stochastic matrix.)