What is the covariance matrix between AX and BY?

Question

Let X = (X1, . . . , Xn) and Y = (Y1, . . . , Yn) be two random vectors (i.e., vectors of random variables). Suppose that the covariance between X_i and Y_j is G_ij . In other words, the covariance matrix between the random vectors X and Y is G. Let A and B be two n × n matrices. What is the covariance matrix between AX and BY?

edit: by linearlity of expectation we know that E(AX) = AE(X) and E(BY) = BE(Y)

Answer 1

I will treat $X$ and $Y$ as column vectors.

$G = \text{Cov}(X, Y) = E[(X-E[X])(Y-E[Y])^\top]$.

\begin{align} \text{Cov}(AX, BY) &= E[(AX-E[AX])(BY - E[BY])^\top] \\ &= E[A(X-E[X])(Y-E[Y])^\top B^\top] \\ &= A E[(X-E[X])(Y-E[Y])^\top] B^\top \\ &= AGB^\top. \end{align}