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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

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$\gamma = \sqrt{\alpha}/||X||_F$ ? –  Raskolnikov Sep 12 '12 at 21:20

1 Answer 1

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 $|\gamma| = \frac{\sqrt{\alpha}}{\|X\|}$.

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