For a random vector X = (X1,..., Xn), we define its covariance matrix to be the n x n matrix C with Cij = E[(X¡ – X;)(X; – X;)], where X; = E[X;]. Show that a covariance matrix C is always positive semi-definite. Can we say that it is always positive definite? hint: An n x n symmetric matrix A is called positive semi-definite if æT Ar > 0 for all r € R". If the inequality holds as a strict inequality for all non-zero x, it is called positive definite.

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