Textbook: Mathematical Methods in Physical Sciences

3. a) Let U be a unitary matrix with eigenvectors [v) and eigenvalues A. Act Uf on the equation U|v) = Alv) to show that UT has the same eigenvectors as U. What are the corresponding eigenvalues? b) Evaluate (Uvv) = (Uv)Tv in two different ways to find a condition on the values 1. c) What can you say about eigenvalues of orthogonal matrices? 4. a) Find an orthonormal basis by the Gram-Schmidt method, starting with the three vectors (1,1,0), (1,1,-1), and (3,0,4). b) Construct orthonormal functions gi(x) and 92(x) on [0,1] from Fi(x) = 1, and F2(x) = e-. What is the "angle" between Fi and F2? c) Write h(x) = 3e- - 2 in terms of your orthonormal set from b)