Let A be a row-stochastic matrix whose associated graph describes an interpersonal influence network (as in the DeGroot model). Let cach individual possess an openness level [0, 1] for i {1,...,n}, describing how open the individual is to changing her initial opinion about a subject; set A diag(A,..., A). Consider the Friedkin-Johnsen model of opinion dynamics - x(k+ 1) = AAx(k) + (In-A)x(0). In this model, a, (k) represents the current opinion of individual and each individual i exhibits an attachment (1-A;) to its initial opinion (0). Consider the following two assumptions: (A1) at least one individual has a strictly positive attachment to its initial opinion, that is, >, < 1 for at least one individual i; (A2) the interpersonal influence network contains directed paths from each individual with open- ness level equal to 1 to an individual with openness level less than 1. Note that, if Assumption (A1) is not satisfied and therefore A = 1 for all i, then we recover the DeGroot opinion dynamics model. In what follows, let Assumption (A1) hold. 1. Show that the matrix AA is convergent if and only if Assumption (A2) holds. (1p) Next, under Assumption (A2), perform the following tasks: 2. Show that the so-called total influence matrix V = (In-AA)-(In-A) is well-defined (i.e. the inverse exists) and row-stochastic. (2p) 3. Show that lima(k) - Vx(0). (1p) Hint: Use the fact that B convergent implies (IB) invertible (as shown in Homework 1).