# Can properties for (circular symmetric) complex random matrices automatically work for real random matrices?

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:

$$$$y = Hx + n,$$$$

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:

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

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!