If X is a random vector
the covariance matrix is
Det(Σ) = ∏ni=1λi≥0 where λi's are eigenvalues of Σ.
Since Σ is symmetric and positive definite, it can be diagonalized and its eigenvalues are all real and positive and the eigenvectors are orthogonal.
det(Σ)=det(VΛVT)=det(V)⋅det(Λ)⋅det(VT)=det(Λ)
det(V)=±1 because det(VV−1)=det(V)det(V−1)=det(V)det(VT)=det(V)2=1
References:
http://www.ece.unm.edu/faculty/bsanthan/EECE-541/covar.pdf
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