Suppose one is given a set of $k$ matrices in $\mathbb{R}^{m \times n}$, $A_1,..,A_k$ and two positive real numbers $\alpha, \beta$.
- When can there exist a matrix $M \in \mathbb{R}^{m \times n}$ s.t $\forall x \in \mathbb{R}^n \text{ and } y \in \mathbb{R}^m$ we have, $$ - \beta y^\top M x \leq y^\top A_i x \leq \alpha y^\top M x , \forall i = 1,\ldots,k$$
?
And sometimes is such a $M$ easy to construct out of the $As$?
$M$ exists only when all the $A_i$s are zero matrices.
If some $A_i$ is nonzero, there will be some $x$ and $y$ such that $y^TA_ix\ne0$. But then the set of inequalities \begin{cases} -\beta y^\top Mx\leq y^\top A_ix\leq\alpha y^\top Mx,\\ -\beta (-y)^\top Mx\leq (-y)^\top A_ix\leq\alpha (-y)^\top Mx \end{cases} cannot be simultaneously satisfied.