Assume that the rank-1 symmetric matrix is $s_ws_w^\top$, where $s_w = [\sin w\ \sin 2w\sin3w\ \cdots\sin Nw]^\top$. And the sum is $S = \sum_{w=0}^T\ s_ws_w^\top$.
What happens when $T$ is fixed and $\Delta w \rightarrow 0$? What is the relationship between the sum and the integral $S = \int_{0}^T s_ws_w^\top\ dw$.
Is there some interesting properties behind the sum/integral of the rank-1 matrices? Or is there some methods to reduce the computation? I have searched for all the morning but nothing.
Could anyone give me some references? Thanks in advance!
EDIT: Sorry for stating the problem wrongly. I want to know what the value of the $\Delta w$ would effect the sum/ integral.