ABSTRACT
We should interpret this as follows. The outer product of the vectors x and y is the matrix.
The notation S=xxt really means S=<xxt>; i.e., the covariance matrix is the average value of the outer product of x with itself. Each sample of the vector x produces a matrix xxt. Since the components of the average of a matrix are just the average of each component, this definition of the covariance matrix agrees with the first one. That is,
This notation for the covariance matrix is too convenient for us not to use it, even though outer vector product is not as familiar as the other two. I shall always use this notation in the context of taking an average value.
