1. An n by n matrix A is said to be a circulant matrix if aij = b(j-i) mod n. For example, three by three matrices are of the circulant if they take the form bo b1 62 A = b2 bo b1 b1 b2 bo (a) Suppose A and F are n by n circulant matrices, show that A + F and AF are both circulant. Moreover, show that AF = FA. (Circulant matrices form a commutative algebra.) (b) Show that there exists a unitary matrix U that diagonalizes circulant matrices. (c) If A is circulant we can associate with it a polynomial n - 1 A( 2 ) = >bizi i=0 which can be thought of as being a finite z-transform of A. (d) Show that if A and F are circulant with T = AF then T(2) = A(2) F(2) if we agree to interpret ed(2) mod (2dim A - 1). (e) A matrix M is an infinitesimally stochastic matrix iff the matrix exponential exp(tM) is a stochastic matrix for all t 2 0. Equivalently, this is saying that matrix M is an intensity matrix (or a transition rate matrix). Show that there exists n by n infinitesimally stochastic matrices whose eigenvalues are {(-1 + ei)}to where {ed} o are the nth roots of 1