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I am having a problem with understanding an English sentence underlined in red below. Can somebody let me understand what it is saying? and what is maximized?

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user122358
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1 Answers1

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The statement is saying two things.

  1. Since for all $x \neq 0$ we have $\frac{(x'd)^2}{x'Bx} \leq d'B^{-1} d$, then also $\displaystyle\max_{x\neq0} \frac{(x'd)^2}{x'Bx} \leq d'B^{-1} d$.

  2. For the particular choice of $x = cB^{-1}d$ (for any $c \neq 0$), we have $\frac{(x'd)^2}{x'Bx} = d'B^{-1} d$, and so $\displaystyle\max_{x\neq0} \frac{(x'd)^2}{x'Bx} \geq d'B^{-1} d$.

These two things put together imply that $$\max_{x\neq0} \frac{(x'd)^2}{x'Bx} = d'B^{-1} d.$$

Yuval Filmus
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  • What does max x mean? I an sorry I still don't get what we are maximizing. I am sorry for asking something really primitive... – user122358 Apr 09 '15 at 05:24
  • It denotes taking the maximum of the quantity to the right over all possible values of $x$ (in this case, all non-zero vectors of the correct format). – Yuval Filmus Apr 09 '15 at 05:24
  • Is it possible to have an example? It is way very abstract to me... If it is too way too much, please don't waste your time. I am still happy with whatever I got from you. – user122358 Apr 09 '15 at 08:46
  • Try taking $B = I$ and see what happens. – Yuval Filmus Apr 09 '15 at 14:32
  • Here's an example of the max notation: $$ \max_{x \in [0,1]} x(1-x) = \frac 14 $$ – Ben Grossmann Apr 09 '15 at 15:37