(a) Let and be the matrices of a linear map on relative to bases and respectively. In this question you may assume without proof that and are similar.
(i) State how the matrix of relative to the basis is constructed from and . Also state how may be used to compute for any .
(ii) Show that and have the same characteristic equation.
(iii) Show that for any the matrices
are similar. [Hint: if is a basis then so is .]
(b) Using the results of (a), or otherwise, prove that any complex matrix with equal eigenvalues is similar to one of
(c) Consider the matrix
Show that there is a real value such that is an orthogonal matrix. Show that is a rotation and find the axis and angle of the rotation.