in MMV-2 is identical to that of MMV-1. Therefore, $\mathbb{X}_{\mu}$ in MMV-1 and MMV-2 are identical in the large system limit.

[**Complex SMV with complex measurement matrix:**]{} The derivations are the same as in MMV-2 above, except that we need to change $\F_{\mu l}$ in  to $\overline{\F}_{\mu l}$  and replace $\T_j$ by $$\label{eq:TransMat}
{\bf T}=\begin{bmatrix}
    0 & 1 \\
    -1 & 0
  \end{bmatrix},$$ because $\overline{\F}_{(\mu+M)l}=\overline{\F}_{\mu l}{\bf T},\ \mu\in\{1,...,M\}$. Using similar steps as above, we obtain that the covariance matrix $\G_{\mu}$ in this case, is also the same as that of MMV-1 with $J=2$.

**Solving $\mathbb{X}_{\mu}$**: For such a structured matrix $\G_{\mu}$ (Fig. \[fig.cov\]), elementary transforms show that the eigen-values (EV’s) are comprised of one EV equal to $\alpha_1=[w_1+(J-1)w_2]+(n-1)[w_3+(J-1)w_4],\ (J-1)$ EV’s equal to $\alpha_2=(w_1-w_2)+(n-1)(w_3-w_4),\ (n-1)$ EV’s equal to $\alpha_3=[w_1+(J-1)w_2]-[w_3+(J-1)w_4]$, and $(J-1)(n-1)$ EV’s equal to $\alpha_4=(w_1-w_2)-(w_3-w_4)$.

Owing to replica symmetry [@Krzakala2012probabilistic; @krzakala2012statistical], we have $m_a=m_b=m$, $Q_a=Q$, and $q_{ab}=q$. Also, in the Bayesian setting, we have $m=q_0=q$ and $Q=\rho$. Thus, $w_2=w_4=0$ ( and ), and $$\label{eq:detSet2}
\begin{split}
\det (\mathbb{I}_{nJ}+&\frac{1}{\Delta}\G_{\mu})= \l(1+\frac{\alpha_1}{\Delta}\r)\l(1+\frac{\alpha_2}{\Delta}\r)^{J-1}\times\\
&\quad \l(1+\frac{\alpha_1}{\Delta}\r)^{n-1}\l(1+\frac{\alpha_1}{\Delta}\r)^{(n-1)(J-1)}\\
&=\l(1+n\frac{w_3}{\Delta+\alpha_4}\r)^J\!\l(1+\frac{1}{\Delta}\alpha_4\r)^{Jn}\!.
\end{split}$$ Considering , we simplify , $$\label{eq:XmuNew}
\lim_{n\rightarrow 0}\mathbb{X}_{\mu}=\operatorname{e}^{-\frac{nJ}{2}\l[\frac{\rho-2m+\Delta+q}{Q-q+\Delta}+\log(Q-q+\Delta)-\log(\Delta)\r]},$$ where we rely on the following Taylor series, $$\label{eq:firstOrder1}
\operatorname{e}^{nk}\approx 1+nk\Rightarrow \operatorname{e}^{-\frac{n}{2}k}\approx (1+nk)^{-1/2},\ n\rightarrow 0.$$

Numerical Results {#sec:numeric}
=================

Given a 