I need to prove the following proposition:
Assume that $H\left( \lambda\right) \in\mathbb{R}^{\left( n-1\right) \times\left( n-1\right) }$ is a matrix function such that
$$H=H^{T}=\frac {1}{2\pi\mathbf{i}}% %TCIMACRO{\toint \nolimits_{C}}% %BeginExpansion {\textstyle\oint\nolimits_{C}} %EndExpansion H\left( \lambda\right) d\lambda \succ 0$$
and let $D_{1} = \mbox{diag}\left( d_{2},\ldots,d_{n}\right)$ and $D_{2} = \mbox{diag} \left( d_{1},\ldots,d_{n-1}\right) $ with $d_{i}>0$. Then,
$$ \frac {1}{2\pi\mathbf{i}}%TCIMACRO{\toint \nolimits_{C}}% %BeginExpansion {\textstyle\oint\nolimits_{C}} %EndExpansion \left( J\left( \lambda\right) ^{-1}D_{1}J\left( \lambda\right) \right) H\left( \lambda\right) \left( J\left( -\lambda^{-1}\right) D_{2}J\left( -\lambda^{-1}\right) ^{-1}\right) \succ 0$$
where $J\left(\lambda\right) \in\mathbb{R}^{n-1 \times n-1}$ is a Jordan block with eigenvalue $\lambda$ and $C$ is a contour including the origin.