Am trying to find a real scalar $\gamma$ such that for a given pair of real rectangular matrices $X,Y$ the following holds:
$\frac{||Y||_{F}^{2}}{5} \leq ||\gamma X||_{F}^{2}\leq ||Y||_{F}^{2}$
Would it be $\gamma=\frac{\sqrt{\alpha}}{||X||_F}$ for any $\alpha$ such that
$\frac{||Y||_{F}^{2}}{5||X||_{F}^{2}} \leq \alpha \leq \frac{||Y||_{F}^{2}}{||X||_{F}^{2}}$ ? Also let me know if there is any condition over $||X||_F,||Y||_F$ that needs to be considered while solving this problem. Thanks.