Question: 1. Use Cauchy's Inequality to show that, for any fixed vector a, the choice b = βa maximizes the quantity |b†a| 2/b†b, for any constant β.
2. Use the definition of the covariance matrix Q to show that Q is Hermitian and that, for any vector y, y†Qy ≥ 0. Therefore, Q is a nonnegative definite matrix and, using its eigenvector decomposition, can be written as Q = CC†, for some invertible square matrix C.