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$x'x=$ $$ \begin{bmatrix} \sum_{i=1}^{n}(X_{1i}-\bar X_1)^2&\sum_{i=1}^{n}(X_{1i}-\bar X_1)(X_{2i}-\bar X_1)\cdots & \sum_{i=1}^{n}(X_{1i}-\bar X_1)(X_{ki}-\bar X_k) \\ \sum_{i=1}^{n}(X_{2i}-\bar X_1)(X_{1i}-\bar X_1)&\sum_{i=1}^{n}(X_{2i}-\bar X_2)^2\cdots \cdots & \sum_{i=1}^{n}(X_{2i}-\bar X_2)(X_{ki}-\bar X_k) \\ \vdots &\vdots&\vdots \\ \sum_{i=1}^{n}(X_{ki}-\bar X_k)(X_{1i}-\bar X_1)&\sum_{i=1}^{n}(X_{ki}-\bar X_k)(X_{2i}-\bar X_2)\cdots & \sum_{i=1}^{n}(X_{ki}-\bar X_k)^2 \\ \end{bmatrix} $$

how can i transform it to correlation matrix as :

$$x'x= \begin{bmatrix} 1 & r_{12}\cdots & r_{1k}\\ r_{21} & 1\cdots & r_{2k} \\ \vdots &\vdots&\vdots \\ r_{k1} & r_{k2}\cdots &1\\ \end{bmatrix} $$?

I know the elements are the numerator of correlation, i.e., $$r_{ij}=\frac{\sum_{u=1}^{n}(X_{iu}-\bar X_i)(X_{ju}-\bar X_j)}{\sqrt{\sum_{u=1}^{n}(X_{iu}-\bar X_i)^2\sum_{u=1}^{n}(X_{ju}-\bar X_j)^2}}$$ .But i can't transform it to correlation formula.

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Let $p_i = [x'x]_{ii}$ be the $i$th diagonal element of $x'x$. Now define the $k\times k$ diagonal matrix $s$ so that $[s]_{ii} = 1/\sqrt{p_i}$ and all non-diagonal elements equal to zero Then I think the correlation matrix is $$ r = s x'x s $$

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  • $\begingroup$ How did you find the way so quickly? $\endgroup$
    – ABC
    Mar 22 '15 at 16:08
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    $\begingroup$ @harry That's a standard result from matrix algebra. $\endgroup$ Mar 22 '15 at 16:12

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