Cross-Covariance matrix from two covariance matrices

Question

Let $x=(x_1,...,x_n)^T$ and $y=(y_1,...,y_n)^T$ be two random vectors, with covariance matrices $E_{xx}$ and $E_{yy}$, respectively.

Could I compute the cross-covariance matrix $E_{xy}$ using $E_{xx}$ and $E_{yy}$?

Regards

Pablo

Answer 1

No! To count the covariance of two random vector you need to know their joint distribution.

Think about simple example. If $x$ and $y$ are independent than you should know that covariance between them is zero. On the other hand if $x=y$ than what is covariance between $x$ and $y$?