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 - Zn T? b) Let V be an n x n unit upper-triangular Toeplitz matrix. Show that Zn - VZ,V-1. 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 Yr of the equation Yr Z, - Z, Y -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 Yr 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 Xp Z, - Z, X, =Y, assuming Y is such that a solution exists. g) Find all solutions X of X Z, - Z, X - Y, assuming Y is such that a solution exists. Hint: You can express X as the sum of the particular solution X, from part f) and a general homogenous solution. h) Given Y, show that the equation X - ZHX Z, -Y always has a unique solution X and give an efficient algorithm to compute X from Y