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Upon reading the Wikipedia article for Covariance I came across the following definition of the cross covariance matrix for random vectors:

For random vectors $X\in\mathbb{R}^m$ and $Y\in\mathbb{R}^n$, the $m\times n$ cross covariance matrix is equal to $$cov(X,Y)=E[(X-E[X])(Y-E[Y])^T]$$ $$=E[XY^T]-E[X]E[Y]^T$$

  1. Is the above definition correct? If so, how does it make sense to multiply $XY^T$, $(X-E[X])(Y-E[Y])^T$ or $E[X]E[Y]^T$ given that they are vectors of different lengths?
  2. From what I understand $cov(X,Y)=C$ where $C$ is a $m\times n$ matrix such that $C_{ij}=cov(X_i,Y_j)$. Is there a (more formal) mathematical expression which is equivalent to this? Or is this exactly how we define $cov(X,Y)$?.
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    $\begingroup$ (1) Did you notice the transpose operator? It tells you that $X$ is treated as an $m\times 1$ matrix and $Y$ as an $n\times 1$ matrix. According to the rules of matrix multiplication, $XY^\prime$ is therefore an $m\times n$ matrix, etc. (2) Please explain what you mean by "more formal": what would be "informal" about the definition you quote? $\endgroup$ – whuber Oct 2 '17 at 17:36
  • $\begingroup$ Thank you this is exactly what I was failing to see! What I meant by formal was "in terms of elementary operations and the expectation operator" (instead of externally defining the matrix entry-by-entry). By what you pointed out I can see that the definition given by Wikipedia is exactly that. Copy paste your comment into an answer if you want, and I'll accept it so you get that rep. $\endgroup$ – Bananin Oct 2 '17 at 18:08

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