scatter versus covariance I am a bit confused about the difference between a scatter matrix (I believe it is sometimes called a scatter matrix) and the covariance. Some distributions, such as elliptical distributions, may not have a covariance matrix but always have a "scatter" matrix. Can the covariance and scatter matrix coincide?

The scatter matrix is a function of a sample set that _estimates_ the covariance matrix. As you point out, there are distributions for which the covariance is undefined, but if you restrict yourself to some finite set of sample points, you can always mechanically form the scatter matrix based on that sample (just as you can always estimate the mean of a 1-variable distribution by taking the average of a sample).

Yes, the two matrices can by luck coincide (assuming the covariance is definable), and in fact, there is a form of the law of large numbers that says that under certain conditions (which I believe are a bit stronger that just the existence of the covariance matrix) the limit for large $n$ of the scatter matrix approaches the covariance matrix.

An interesting exercise is to find a distribution for which the covariance matrix exists but is not accurately estimated by the limit of scatter matrices. As I said, I believe there are some, but I can't think of an example.