Let $M_1$ and $M_2$ be matrices of dimension $2n_1\times 2n_1$ and $2n_2\times 2n_2$ respectively; Let $P_1>0$ and $P_2>0$ be the positive definite solutions of the following equations
$$ \left[\begin{array}{cc} I_{n_1} & -P_{1}\end{array}\right]M_{1}\left[\begin{array}{c} P_{1}\\ I_{n_1} \end{array}\right] =0$$ and
$$ \left[\begin{array}{cc} I_{n_2} & -P_{2}\end{array}\right]M_{2}\left[\begin{array}{c} P_{2}\\ I_{n_2} \end{array}\right] =0$$ what is the condition between $M_1$ and $M_2$ which involves $$\text{Trace}(P_1)<\text{Trace}(P_2)$$
