(Solved): 7. The covariance matrix of a random vector is always positive semidefinite. ...

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7. The covariance matrix of a random vector is always positive semidefinite. Let $$\mathcal{C}$$ be the covariance matrix of the random vector $$X=\left(X_1,\dots ,X_n\right)$$ (not necessarily Gaussian). Show that $$\mathcal{C}$$ is always positive semidefinite; i.e., $$\underset{i,k=1}{\overset{n}{?}}{a}_i{a}_j{e}_{ij}?0,\text{ for any }{a}_1,\dots ,a_n?\mathbb{R}.$$ Exercises 47 In particular, show that it is positive definite if and only if $$X$$ is nondegenerate. Hint: Write the left side as the variance of some random variable.