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I am extracting information of my problem from proving an application problem, if there is anything not clear, please let me know. Suppose we parameterize a structured matrix in following form \begin{align} \label{matrix:1} \tag{$\star$} \begin{pmatrix} 0 & -a_1 & 0 & 0 & -b_1 \\ 1 & -a_2 & 0 & 0 &