Exercise 1.3 Show that Cov(Y) is nonnegative definite for any random vector Y.

The covariance of two random vectors with possibly different dimensions can be defined. If Wr×1 and Ys×1 are random vectors with EW = γ and EY = μ, then the covariance ofW and Y is the r×s matrix Cov(W,Y) = E[(W −γ )(Y −μ)].

In particular, Cov(Y,Y) = Cov(Y). If A and B are fixed matrices, the results of Exercise 1.2 quickly yield Cov(AW,BY) = ACov(W,Y)B