What is the difference between the sum of two covariance matrices and the covariance matrix of the sum of two variables? I'm wondering if someone could help to explain the difference between two covariance matrices. Suppose that ${\bf K}_X$ and ${\bf K}_Y$ are two covariance matrices of real random vectors. 
What is the difference between ${\bf K}_X+{\bf K}_Y$ and ${\bf K}_{X+Y}$?
 A: While searching through unanswered questions I noticed this one again and decided, in agreement with whuber, that keeping essentially answered questions off of the unanswered tab is higher priority than my own personal preferences about what is "worthy" of answer vs. comment status, so I pasted my comment as an answer.
They are different because ${\bf K}_{X} + {\bf K}_Y$ is the sum of two covariance matrices while ${\bf K}_{X+Y}$ is the covariance matrix of the random variable $X+Y$. To see why the two matrices are different, use the bilinearity of covariance to see that
$$ [{\bf K}_{X+Y}]_{ij}=[{\bf K}_{X}]_{ij} +[{\bf K}_{Y}]_{ij}+ {\rm cov}(X_i,Y_j)+{\rm cov}(X_j,Y_i)$$
i.e. the cross-covariances are missing from ${\bf K}_{X} + {\bf K}_Y$ (note I assume $X,Y$ are of equal dimension to ensure that question makes sense). So, ${\bf K}_{X+Y}$ is the covariance matrix of $X+Y$ and ${\bf K}_{X} + {\bf K}_Y$ represents the special case where ${\rm cov}(X_i,Y_j)=-{\rm cov}(X_j,Y_i)$ for each pair $(i,j)$, the most notable example being when every element of $X$ is uncorrelated with every element of $Y$. 
