Given $a_1+a_2+\cdots+a_n=0$ and $a_i^2+b_i^2=1,\forall i\in\{1,2,\cdots,n\}$, prove the following matrix is negative semi-definite
$$\left[\begin{array}{ccccc|c} -b_1&0&\cdots&0&0&b_1\\ 0&-b_2&\cdots&0&0&b_2\\ \vdots&\ddots&\ddots&\vdots&\vdots&\vdots\\ 0&0&\cdots&-b_{n-1}&0&b_{n-1}\\ 0&0&\cdots&0&-b_n &b_n\\ \hline b_1 &b_2 &\cdots&b_{n-1}&b_n &-b_1-b_2-\cdots -b_n \end{array}\right]$$