Consider the product, H, that is the product of an arbitrary vector, a, and the covariance vector, x:
(1) |
We can re-arrange the above by moving the constant vector a inside expectation operator so that we have,
(2) |
If we define
(3) |
H = E[ Y^{T} Y] | (4) |
This ``squared'' quantity is clearly never negative, so that we can conclude that the covariance matrix is non-negative definite.