        \begin{align}
        \label{equ:case2-decomp}
        \|L^{\top} Ue_k\|^2=\sum_{i=1}^r \widehat{s}_i^2=-\frac{b^2 \widehat{s}_1^2}{1-b^2}+\frac{\widehat{s}_1^2}{1-b^2}+\sum_{i=2}^r \widehat{s}_i^2 \end{align}
