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I have a $N \times N$ covariance matrix $C$ of a multivariate Normal distribution. Can any of the elements of the Covariance matrix $C$ be negative for a real-valued distributions ?

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Any negative correlation between two elements will end up with a corresponding negative entry in the covariance matrix.

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Yes, for example $$ \begin{pmatrix} a+b & a-b \\ a-b & a+b \end{pmatrix} $$ can appear as covariance matrix for any positive eigenvalues $2a$, $2b$.

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