I have an ODE system $Ly'= Ry$ (initial condition $y(0)=c$), where $L=\begin{pmatrix} 1 &0& 0& 0& 0& 0& 0& 0\\ 0 &0& 0& 0& 0& 0& 0& 0\\ 0 &0& -1& 0& 0& 0& 0& 0\\ 0 &0& 0& 0& 0& 0& 0& 0\\ 0 &0& 0& 0& 1& 0& 0& 0\\ 0 &0& 0& 0& 0& 0& 0& 0\\ 0 &0& 0& 0& 0& 0& -1& 0\\ 0 &0& 0& 0& 0& 0& 0& 0 \end{pmatrix}$ and $R=\begin{pmatrix} -1 &0.25& 0.25& 0.25& 0& 0& 0& 0\\ 0.25 &-1& 0.25& 0.25& 0& 0& 0& 0\\ 0.25 &0.25& -1& 0.25& 0& 0& 0& 0\\ 0.25 &0.25& 0.25& -1& 0& 0& 0& 0\\ 0.25 &0& 0& 0& 0& 0& 0& 0\\ 0 &0.25& 0& 0& 0& 0& 0& 0\\ 0 &0& 0.25& 0& 0& 0& 0& 0\\ 0 &0& 0& 0.25& 0& 0& 0& 0 \end{pmatrix}$ and $y(x)$ is a vector of length 8.
That is, the matrix $L$ has a few zero rows. If it were non-singular, I would solve $(L^{-1}R)y=\lambda y$ and expand $y$ in terms of the eigenvectors $v_n$ ie $y=\sum_{n=1}^8a_ne^{\lambda_nx}v_n$ (assuming no repeated eigenvalues), and use the initial condition to find the $a_n$. Does anyone know a way to modify this procedure when $L$ is singular such as in my case?