$K=\{1,2,\dots,m\}$
$A_1,\dots, A_m$ be any square matrices. $n\times n$ say.
If there exist positive definite symmetric matrices $P_1,\dots, P_m$, $(n\times n)$ say.
such that, $A_i^TP_jA_i-P_i<0\forall (i,j)\in K\times K$ then I need to prove that $\exists\epsilon\in (0,1)$ such that
$A_i^TP_jA_i<(1-\epsilon)^2P_i$ holds for $(i,j)\in K\times K$
Thanks for helping.
For fixed $i,j$, consider the function $f:\mathbb R\to M_n(\mathbb R)$: $f(\epsilon)=(1-\epsilon)^2 P_i-A_i^T P_j A_i$. This map is continuous, and since $f(0)$ is assumed to be symmetric positive definite, and the set of symmetric positive definite matrices is open in $M_n(\mathbb R)$, the set of $\epsilon$ such that $f(\epsilon)$ is symmetric positive definite is a neighborhood of $0$. Call that neighborhood $N_{i,j}$. The intersection of the $K\times K$ neighborhoods $N_{i,j}$ of $0$ is another neighborhood of $0$. The intersection $N\cap (0,1)$ is non-empty, and for all $\epsilon$ in it, all your matrix inequalities are satisfied.