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EDIT2: Suppose we relax the family of possible matrices, to matrices $\{P_i\}$ for which there exists nonsingular $Q_{ij}$ such that $Q_{ij}^TP_iQ_{ij}=P_j$ for all pairs $(i,j)$. Or even that $P_i = Q_i^TA_0Q_i$ (which are part of the assumptions of my original problem). Would this help somehow?

EDIT2: Suppose we relax the family of possible matrices, to matrices $\{P_i\}$ for which there exists nonsingular $Q_{ij}$ such that $Q_{ij}^TP_iQ_{ij}=P_j$ for all pairs $(i,j)$. Would this help somehow?

EDIT2: Suppose we relax the family of possible matrices, to matrices $\{P_i\}$ for which there exists nonsingular $Q_{ij}$ such that $Q_{ij}^TP_iQ_{ij}=P_j$ for all pairs $(i,j)$. Or even that $P_i = Q_i^TA_0Q_i$ (which are part of the assumptions of my original problem). Would this help somehow?

More context: This question arises since I'm looking for a "special" point $x^*\in E$ (say optimal in the sense of an objective $x^TA_0x$) and I want to make sure (or at least almost sure) that $x^*$ is regular.

More context: This question arises since I'm looking for a "special" point $x^*\in E$ (say optimal in the sense of an objective $x^TA_0x$ with $A_0$ positive definite) and I want to make sure (or at least almost sure) that $x^*$ is regular.