10. Eigentheory of Rotation Matrices: The objective of this Exercise is to show that a rotation matrix: cos(0) -sin(0) Re = sin(0) cos(0) does not have any real eigenvalues, for any value of 0 except when 0 = лn, where n is an integer. a. b. C. Find the characteristic polynomial of this matrix. Show that the discriminant of the characteristic equation is negative, unless 0 = n for some integer n. If so, show that there are exactly two different matrices Re, and find their eigenvalues. Now, argue geometrically that the rotation matrix can only have real eigenvalues if Ꮎ = лn for some integer n. Hint: draw the effect of Re on an eigenvector. We will further study the eigentheory of rotation matrices when we introduce imaginary eigenvalues and eigenvectors in Chapter 10.