Prove a theorem about an upper bound for entropy of a random vector

Question

There is a theorem that:

if Z is any zero-mean, complex random vector with covariance \$E[ZZ^H]=R_z\$, then \$H(Z)\leq \log|{\pi eR_z}|\$, with equality holding if and only if Z has a circularly symmetric Gaussian distribution and \$H(Z)\$ is the entropy of Z.

I do not know what the name of this theorem to seek its proof in books or articles. I will be thankful if someone tell its name to me, or offers a proof for this theorem.

Answer 1

This theorem has not a specific name, but a proof for this has been offered in the following article on page 4:

http://lthiwww.epfl.ch/~leveque/Projects/telatar.pdf