Question

Consider the following hermitian matrix a) Calculate the trace and the determinant of this matrix. b) Find the eigenvalues and compare their product and sum to the determinant and trace respectively. (It is a general result for any matrix that can be diagonalized, that the trace of a matrix is equal to the sum of its eigenvalues and that the determinant of a diagonalizable matrix is equal to the product of its eigenvalues. If these conditions are satisfied, you can be sure that the matrix can be diagonalized before or without carrying out the diagonalization procedure) c) Write down the diagonalized form of T based on what you know about the eigenvalues alone! d) Find the eigenvectors of the matrix. Within the degenerate sector, construct two linearly independent eigenvectors (it is this step that is always possible for a Hermitian matrix, but not for an arbitrary matrix -- see problem 2) Orthogonalize them and check that both are orthogonal to the third (non-degenerate) vector. Normalize all three eigenvectors e) Construct the unitary matrix S that diagonalizes T, and show explicity that the similarity transform reduces T to the expected diagonal form

Answer 1

"Given that T^- = (2 i 1 -i 2 i 1 -i 2) a)Trace = 2 + 2 + 2 = 6