Circular symmetric complex Gaussian zero mean PDF is defined as :
$$f(z)= \frac{1}{\pi^N\Vert{M}\Vert} e^{-z^*M^{-1}z} $$ where $M$ is hermitian semi positive definite, $z \in \mathbb{C}^{N \times 1}$ and $\Vert\cdot\Vert$ is determinant operation.
Question
If we consider the case of one dimensional vector $z$ containing one real number (i.e., $z=z+0i$), then it should be defined by above relation, but instead it turns out to be $\frac{e^{-\frac{z^2}{\sigma ^2}}}{\pi \sigma ^2}$, from above definition.
The confusion is why it is not $\frac{e^{-\frac{z^2}{2 \sigma ^2}}}{\sqrt{2 \pi } \sigma }$? What am I missing to understand.
Recall that you have a random complex $n$-vector $Z$ and the corresponding real random $2n$-vector $V$, the dimensions are different. On the one hand you have the real density probability: $$ f_V(v)= \frac{1}{(2\pi)^N\sqrt{||K_V||}} e^{\Large-\frac{v^TK_V^{-1}v}{2}} $$ on the other hand you have the complex density probability: $$ f_Z(z)= \frac{1}{\pi^N||K_Z||} e^{-z^*K_Z^{-1}z}. $$ The relation between these densities is explained here (please read section $3$ and section $4$, especially $(10)$-$(13)$).