quadratic eigenvalue problem reformulation with eigenvalues on unit cycle

Question

Suppose a quadratic eigenvalue problem is $$(A_2\lambda^2+A_1\lambda+A_0)x=0$$ where $\lambda$ is the eigenvalue and $x$ is the eigenvector. the quadratic form can be rewritten by $$\left[ {\begin{array}{*{20}{c}} 0&I\\ {{A_0}}&{{A_1}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} x\\ {\lambda x} \end{array}} \right] = \lambda \left[ {\begin{array}{*{20}{c}} I&0\\ 0&{ - {A_2}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} x\\ {\lambda x} \end{array}} \right]$$ Now, suppose $\lambda$ is restricted to eigenvalues on unit circles,i.e.,$\lambda = \frac{{1 + i\sigma }}{{1 - i\sigma }}$. I want to further change the equation above as $$\left[ {\begin{array}{*{20}{c}} { - I}&I\\ {{X_1}}&{{X_2}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} x\\ {\lambda x} \end{array}} \right] = i\sigma \left[ {\begin{array}{*{20}{c}} I&I\\ {{X_3}}&{{X_4}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} x\\ {\lambda x} \end{array}} \right]$$ What should $X_{1,2,3,4}$ be?