Let $X, Y$ be random vectors. Let $K = \text{Cov}(X,Y) = E[XY^T] - E[X]E[Y]^T$ be the covariance matrix of $X$ and $Y$. Assume $K$ is low rank.
I'm trying to come up with simple intuitive examples about what having a low rank covariance matrix means but I'm having trouble. I understand that a low rank matrix means most of the column vectors are linearly dependent on other column vectors, and I understand that the covariance matrix shows the variance relationships between each random variable. But from here I'm having trouble coming up with an intuitive explanation or example of where this would be useful.