Let $\boldsymbol{R}(u,v);~u,v\in\mathbb{R}$ be a Hermitian matrix (of Hermitian functions) with entries \begin{equation} r_{ij}(u,v) = 1 + Ae^{-2\pi i \phi_{ij}(ul_0 + vm_0)}; A\in\mathbb{R},l_0\in\mathbb{R},m_0\in\mathbb{R},\phi_{ij}\in\mathbb{Z}. \end{equation} Furthermore let, $0<A<1$, $l_0 \neq 0, m_0 \neq 0$, $\phi_{i,i}=0$ (diagonal), $\phi_{ij} \neq 0$ (non-diagonal), $\phi_{ij} > 0;~\forall j>i$, $\phi_{ij} = -\phi_{ji}$ and gcd($\{\phi_{ij}\}_{j>i}$) $=1$. If $\boldsymbol{G}(u,v)=\lambda(u,v)\mathbf{x}(u,v)\mathbf{x}^{H}(u,v)$, where $\lambda(u,v)$ is the largest eigenvalue of $\boldsymbol{R}(u,v)$, $\mathbf{x}(u,v)$ is its associated eigenvector (normalized) (of $\lambda(u,v)$) and $()^H$ is the Hermitian transpose then prove that the entries $g_{ij}(u,v)$ of $\boldsymbol{G}(u,v)$ are Hermitian functions that are periodic in the $u$ and $v$ direction with periods respectively equal to $\frac{1}{|l_0|}$ and $\frac{1}{|m_0|}$.

I arrived at this problem by studying ghost sources (radio interferometry) and alternating least squares calibration. I have a few final comments (based on experimental observation). The $\boldsymbol{R}(u,v)$ matrix seems to be rank 2 (how to prove this though?). Can an analytic expression for $\lambda(u,v)$ be derived (for any dimension of $\boldsymbol{R}(u,v)$) and can it aid in the proof? Also it seems that $\lambda(u,v)$ is always positive (how to prove this)? Once we have $\lambda(u,v)$? How do we derive the periodicity of $\mathbf{x}(u,v)$? Can anyone suggest an approach to derive this proof?