I was watching a lecture on Gaussian Process and when the covariance matrix was introduced, the tutor [explained][1] that the matrix is $(n \times n)$ because every point is covered twice - we include the information about $cov(x_1, x_2)$ and about $cov(x_2, x_1)$. She then said: "that's because your covariance can vary in different directions".

From the definition of the covariance I see no reason for these to be different: $cov(X, Y) = E[(X - E(X))(Y - E(Y)]$.

How is it possible that the $cov(x_1, x_2) \neq cov(x_2, x_1)$ inside the covariance matrix? Could you give me an example of when that could be the case?


  [1]: https://www.youtube.com/watch?v=UpsV1y6wMQ8&feature=youtu.be&t=1130