a) Let T be an n x n Toeplitz matrix. Let Z,, denote the n x n shift-up matrix. What is the maximum rank of the matrix TZn - ZnT? b) Let V be an n x n unit upper-triangular Toeplitz matrix. Show that Z. = VZ,VI c) Find all solutions XR of the equation XR Zn - Zn XR=0. What is the dimension of the solution space? d) Find all solutions Yz of the equation Yr Z. - Z. Yo =0. What is the dimension of the solution space? e) Consider the equation X Zn - Z, X = Y. Show that this equation has a solution X, if and only if, trace(YZ Y) =0, where Yz is from part d). Hint: How many linearly independent constraints on Y does this give? How does that compare to the dimension of the null-space? What do you know about the angle between the range space and left null-space of a linear operator? f) Give an efficient algorithm to find a particular solution X, such that X, Zn - Z, Xp = Y, assuming Y is such that a solution exists. g) Find all solutions X of XZn - Zn X = Y, assuming Y is such that a solution exists. Hint: You can express X as the sum of the particular solution Xp from part f) and a general homogenous solution. h) Given Y, show that the equation X - Z, X Zn - Y always has a unique solution X and give an efficient algorithm to compute X from Y