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 $\mathbf{r}$):
$$ Corr(\hat{s},r_k) = \frac{\langle \hat{s}r_k \rangle - \langle \hat{s} \rangle \langle r_k \rangle}{\sqrt{( \langle \hat{s}^2 \rangle - \langle \hat{s} \rangle^2) }\sqrt{( \langle r_k^2 \rangle - \langle r_k \rangle^2) }} = \frac{(\Sigma \mathbf{w})_k}{\sqrt{(\Sigma_{kk}\mathbf{w}^T \Sigma \mathbf{w})}} $$
Thanks!!
