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I am dealing with a theorem which relates to circularly symmetric complex Gaussian random matrices (CSGRM).

It seems quite tempting to assume that the theorem also extends to real-valued Gaussian random matrices, since CSGRMs have their real and imaginary parts completely uncorrelated, I can essentially "assume" the imaginary part to be zero.

In particular I am dealing with the following multi variate Gaussian model:

\begin{equation} y = Hx + n, \end{equation}

where $H$ is $r\times t$ complex valued, $x$ is $t$ complex valued, $y$ is $r$ complex valued. Theorem 1 of the paper "Capacity of Multi‐antenna Gaussian Channels" (available: http://web.mit.edu/18.325/www/telatar_capacity.pdf ) has Theorem 1 as:

\begin{equation} \mathbb{E}[\log \text{det} (I_r +(P/t)HH^{\dagger})] \end{equation}

which the author claims is true assuming $x$ is a CSGR vector.

Question

Can I readily just assume it also naturally extends to real Gaussian random matrices? Based on the few random matrix properties Theorem 1 is based on, it seems like it should ... but I dont have the confidence to say this completely!

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