Using the standard method explained in Chap. 1, define the eigen values λ1 = 3 and λ2 = −1 for the matrix P and the eigen vectors √ 1/√2|(1, 1)| and 1/√2|(1, −1)|. For the matrix Q, the eigen values are λ1 = 5 and λ2 = 1, the eigen vectors being 1/√2|(1, 1)| and 1/√2|(1, −1)|. Thus the eigen vectors for the commutating matrices are identical.