7. [8 points, 4 points each] Given a matrix A and a unit vector v. J is called an eigenvector of A with complex eigenvalue X if it has the following property: AU = Xv. That is, multiplying vector v by the matrix A results in the vector v multiplied by a complex number X. (a) Prove that if A is a unitary matrix, then the eigenvalue \\ must have the form e*\" for some angle 6. (hint: consider the inner product of Av with itself.) (b) Prove that if A is hermitian, the eigenvalue A must be real. (hint: compare the number ve (Av) and (Av) ev.) 8. [8 points, 4 points each] Let H be any hermitian matrix and U be any unitary matrix. (a) Show that UHUt is also hermitian (b) Show that (UHU')" =UH"U