Was wondering on what would be the real number (scalar) $\gamma$ that needs to be multiplied with each entry in a real rectangular matrix $X_{m\times n}$ such that the Frobenius norm of $X$ equals a given positive value $\alpha$?
i.e, Find a $\gamma$ such that $||\gamma X||_{F}^{2}=\alpha$.
Am expecting that $\gamma$ would be a function of $mn$ and $\alpha$.
Let me know, if there is some condition or notation that I might have left out.
Was posting a question based on an inequality based on the Frobenius norm, but it required this question to be answered to get my notation right in the other question. Thanks
Best Answer
It is a norm, so $\|\gamma X \|^2 = |\gamma|^2 \|X\|^2$. Consequently, if you want $\|\gamma X \|^2 = \alpha$, you must choose $\gamma$ to satisfy $|\gamma| = \frac{\sqrt{\alpha}}{\|X\|}$.