1. Below, for x = ()), ||*|| stands for the 2-norm (z + x)1/2. (a) Suppose a 2 x 2 matrix takes the form A = [-%] for some real numbers a,b. Show that the eigenvalues of A are a ib. (b) Show that if y = Ax with A as in (a), then || y ||2 = (a? +62)|||||2. (c) Fixing a step At > 0, write the Crank-Nicolson iteration approximating the initial value problem (9)' = [-91 ] (%) with (*) (0) = () (the true solution rotates clockwise through the circle of cen- ter 0 and radius 1). That is to say, find a matrix (At) approximating A(At) = { cst cons(AH! such that (t) = (81) 5) Un+1 (d) Use (a)-(b) to show that for every n > 0, um+1 + vm+1 = u + vm. This means that the Crank- Nicolson solution remains on the unit circle, like the true solution. (e) When the true solution rotates by an angle At on the circle, what is the angle of rotation of the Crank-Nicolson solution ? How many iterations will it take for the true solution and the Crank-Nicolson one to be approximately diametrically opposed on the circle ? 1. Below, for x = ()), ||*|| stands for the 2-norm (z + x)1/2. (a) Suppose a 2 x 2 matrix takes the form A = [-%] for some real numbers a,b. Show that the eigenvalues of A are a ib. (b) Show that if y = Ax with A as in (a), then || y ||2 = (a? +62)|||||2. (c) Fixing a step At > 0, write the Crank-Nicolson iteration approximating the initial value problem (9)' = [-91 ] (%) with (*) (0) = () (the true solution rotates clockwise through the circle of cen- ter 0 and radius 1). That is to say, find a matrix (At) approximating A(At) = { cst cons(AH! such that (t) = (81) 5) Un+1 (d) Use (a)-(b) to show that for every n > 0, um+1 + vm+1 = u + vm. This means that the Crank- Nicolson solution remains on the unit circle, like the true solution. (e) When the true solution rotates by an angle At on the circle, what is the angle of rotation of the Crank-Nicolson solution ? How many iterations will it take for the true solution and the Crank-Nicolson one to be approximately diametrically opposed on the circle