I have the given Linear matrix inequality
$$ {X^T}{\left( {{P^{ - 1}} - {\gamma ^{ - 2}}}I \right)^{ - 1}}X \succeq {\left( {X - C} \right)^T}{\left( {{T^{ - 1}} - {\gamma ^{ - 2}}}I \right)^{ - 1}}\left( {X - C} \right) - {\gamma ^2}{C^T}C $$
The matrices involved have following properties:
- Both matrices $P$ and $T$ are positive definite and $T \succ P$. These two matrices were obtained by solving other two LMIs I am not mentioning here (those LMIs also depend on $X$ and $C$).
- $\gamma$ is a positive scalar and is a variable here.
- All matrices are square. The matrices $X$ and $C$ are arbitrary and not necessarily symmetric let alone PSD or PD.
Problem: How do I go about showing the feasibility of the LMI for any $X$ and $C$ by selecting some $\gamma$?
Attempts: I have tried several algebraic manipulations but to no use. Note that in the case $C^TC$ is PD (arise when $C$ has full rank), it can be easily made feasible by selecting $\gamma$ large so that the second term in right hand side dominates the first and also by noting that the squeezed terms which depends on $\gamma$ decrease (${\left( {{T^{ - 1}} - {\gamma ^{ - 2}}} I\right)^{ - 1}}$ and ${\left( {{P^{ - 1}} - {\gamma ^{ - 2}}}I \right)^{ - 1}}$) , but unfortunately that's not the case and best we could say is that $C^TC$ is only PSD. Some of you may say why not writing it in Schur complement form and solve SDP problem, but I am trying to prove feasibility, I am not interested in solving the problem. I am happy to answer any questions. I appreciate any tips or advice. Thanks!