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This might be silly but how would I solve the following equation for $x$?

$$ \left\| Y - \frac{Z_i}{x} \right\|^2_2 = 2t $$

Rodrigo de Azevedo
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John
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  • Assuming $Y$ and $Z_i$ are simply vectors in $\mathbb{R}^n$, your equation reads $\sum_{k=1}^n (Y_k - (Z_i)_k/x)^2 = 2t$, which is simply a quadratic equation. – filmor Oct 28 '13 at 14:52
  • @filmor I see what you mean - actually, I am trying to minimize this problem : http://math.stackexchange.com/questions/542907/minimization-of-norms – John Oct 28 '13 at 15:11
  • @filmor I am not entirely sure how to do reach that conclusion by just taking the derivative and setting to 0. – John Oct 28 '13 at 15:13
  • Use the inner product definition of the $l_2$ norm, and set the derivative equal to 0. – AAP Oct 28 '13 at 19:04
  • I see. Thank-you. – John Oct 29 '13 at 14:14
  • What are $Y, Z_i,x$ and $t$?. Note let $y\in\mathbb{R}^n$, then $| y|^2_2= y^Ty$. – CroCo Sep 03 '23 at 22:29

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Write squared norm as inner product: $$\| Y - Z_i/x\|^2 = \|Y\|^2 - \frac{2}{x} \langle Y, Z_i\rangle +\frac{1}{x^2}\|Z_i\|^2 $$ Equating this to $2t$ yields a quadratic equation for $1/x$. (Or a quadratic equation for $x$, if you multiply everything by $x^2$).

Post No Bulls
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