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I assume you mean covariance matrix and I assume that you are familiar with the definition:
C = E[(X-u)(X-u)T]
where X is a random vector and u = E(X) is the mean
The definition of non-negative definite is:
xTCx ≥ 0 for any vector x Є R
So is xTE[(X-u)(X-u)T]x ≥ 0?
Then, from one of the covariance properties:
E[(xT(X-u))((X-u)Tx)] = E[xT([(X-u)(X-u)T]x)] = E[((xTI)x)] = E[xTx]
Finally, since we've already defined x to have only real values, xTx is therefore non-negative definite by definition.
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