Interpretation of $E((X-E(X))(Y-E(Y))^T)$?

Question

In the single dimensional case, it is the case that $E((X-E(X))(Y-E(Y)))$ corresponds to the covariance between X and Y. However, in the multidimensional setting, the covariance matrix of the multivariate random variable W is defined by $E((X-E(X))(X-E(X)^T))$.

I am therefore curious about whether or not there is an interpretation of $E((X-E(X))(E(Y-E(Y))^T)$, as this seems to be a multidimensional analogue of the covariance formula, however, this does not correspond to the covariance matrix of a random variable.

Is the resultant matrix used in any contexts, and if so what is its interpretation?

Answer 1

I am therefore curious about whether or not there is an interpretation of $\Bbb E((X-\Bbb E(X))(Y-\Bbb E(Y))^\intercal)$, as this seems to be a multidimensional analogue of the covariance formula, however, this does not correspond to the covariance matrix of a random variable.

This is called the joint covariance matrix for the random vectors. It is the matrix of covariances of members of each random vector.

$${[\mathsf{Cov}(X,Y)]}_{i, j}=\mathsf{Cov}(X_i,Y_j)$$

(it is sometimes also referred to as the cross-covariance matrix).