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Pearson correlation of neural responses with it's linear estimation


I am trying to anderstand the following fact from this article (page 13): How can single neurons predict behavior

Suppose I have a linear estimation of a stimulus: $ hat{s} = mathbf{w}^T(mathbf{r} - mathbf{f}(s_0)) + s_0$

where $mathbf{w}$ is a vector of weights, $mathbf{r}$ is a vector of responses responses of two neurons to a stimulus, $mathbf{f}$ is the vector of average neural responses to the stimulus, and the stimuli (angels between $-pi$ to $pi$) are symetric (amount and location) around $s_0 = 0$.

Can anyone see why the following is true for the Pearson correlation (where $Sigma$ is the covariance matrix of $mathbf{r}$):

$$ Corr(hat{s},r_k) = frac{langle hat{s}r_k angle - langle hat{s} angle langle r_k angle}{sqrt{( langle hat{s}^2 angle - langle hat{s} angle^2) }sqrt{( langle r_k^2 angle - langle r_k angle^2) }} = frac{(Sigma mathbf{w})_k}{sqrt{(Sigma_{kk}mathbf{w}^T Sigma mathbf{w})}} $$

Thanks!!