(i) Find the eigenvalues and eigenvectors of the following matrices and show that both are diagonalisable:
(ii) Show that, if two real matrices can both be diagonalised using the same basis transformation, then they commute.
(iii) Suppose now that two real matrices and commute and that has distinct eigenvalues. Show that for any eigenvector of the vector is a scalar multiple of . Deduce that there exists a common basis transformation that diagonalises both matrices.
(iv) Show that and satisfy the conditions in (iii) and find a matrix such that both of the matrices and are diagonal.