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!!