1. Let H be a general Hermitian matrix with TrH = 0 and detH = −1. (a) Show that any such a matrix can be represented as
Let H be a general Hermitian matrix with TrH = 0 and detH = −1.
(a) Show that any such a matrix can be represented as
(b) Show that H is a unitary matrix.
(c) Explain, without a calculation, what the eigenvalues of H should be.
(d) Since n = nxi + nyj + nxk is a unit vector, it is convenient to represent its components in spherical polar coordinates (θ, φ). Show that the matrix H in this representation is
(e) Diagonalise H with a unitary transform, Λ = U†HU, choosing the transformation matrix U to be of the same form as H. Given θ and φ in H, fifind the corresponding values of parameters in U. Verify your answer.
2.
Consider the following matrix:
(a) Find its eigenvalues and eigenvectors.
(b) Diagonalise A with a similarity transform, Λ = P−1AP.
(c) Use the diagonalisation to calculate an arbitrary odd power A2n−1 (where n = 1, 2, · · · ). Check that your answers reproduces matrix A when n = 1.
(d) Find a square root of A, i.e. any one matrix M such that M2 = A. Verify your answer by explicitly calculating matrix elements {M2}ij and showing your work in detail.
3.
Transverse oscillations of an infinite elastic string are described by the displacement amplitude u(x, t), which obeys the wave equation