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I'm trying to understand the theoretical reasoning behind the method, but I can't understand a particular step in the middle of this page.

"The constraint on the numbers in $v_1$ is that the sum of the squares of the coefficients equals $1$. Expressed mathematically, we wish to maximize $$\frac1N\sum_{i=1}^NY_{1i}^2$$

where $$y_{1i}=v_1'z_i,$$

and $$v_1'v_1=1$$

(this is called "normalizing" $v_1$). Computation of first principal component from $R$ and $v_1$. Substituting the middle equation in the first yields

$$\frac1N\sum_{i=1}^NY_{1i}^2=v_1'Rv_1."$$

I don't understand how $R$ suddenly appeared in this equation. The right hand side "$v_1' R v_1$" seems to have appeared out of nowhere.

Please help

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1 Answer 1

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\begin{align} \frac1N\sum_{i=1}^N Y_{1i}^2 &= \frac1N\sum_{i=1}^N (v_1'z_i)(z_i'v_1)\\ &=v_1'\left(\frac1N \sum_{i=1}^N z_iz_i' \right) v_1 \\ &= v_1'Rv_1 \end{align}

where $R=\frac1N \sum_{i=1}^N z_iz_i'.$

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  • $\begingroup$ umm, question. why can you alter the order of the two variables when you square them? in v1'zi * zi'v1 $\endgroup$ Commented May 4, 2018 at 1:33
  • $\begingroup$ To make it into a form that is desirable to us. $\endgroup$ Commented May 4, 2018 at 1:34
  • $\begingroup$ sorry I can't really understand. am I confused about what the (') mark means? i thought maybe that it was referring to a matrix transpose since derivative doesn't make sense here $\endgroup$ Commented May 4, 2018 at 1:37
  • $\begingroup$ we have $(v_1'z_i)=(z_i'v_1)$ since they are scalar. yes ' means transpose. $\endgroup$ Commented May 4, 2018 at 1:38
  • $\begingroup$ ah ok then it makes sense if they are scalars.. is R = 1/N sum zi * zi' the definition of correlation? or is there some reason why that is the case? $\endgroup$ Commented May 4, 2018 at 1:42

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