The influence of eigendecomposition on the periodicity of a (rank 2) Hermitian matrix (of functions).

The influence of eigendecomposition on the periodicity of a (rank 2)
Hermitian matrix (of functions).

pLet $\boldsymbol{R}(u,v);~u,v\in\mathbb{R}$ 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, $0lt;Alt;1$, $l_0 \neq 0, m_0 \neq 0$,
$\phi_{i,i}=0$ (diagonal), $\phi_{ij} \neq 0$ (non-diagonal), $\phi_{ij}
gt; 0;~\forall jgt;i$, $\phi_{ij} = -\phi_{ji}$ and
gcd($\{\phi_{ij}\}_{jgt;i}$) $=1$. 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 $\frac{1}{|m_0|}$./p pI 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?/p