The eigenvalue problem for a matrix of a special form and its applications (Q1362647)

The eigenvalue and eigenvectors of the matrix \(A=\left[\begin{smallmatrix} 0 & I\\ C^{-1}R & C^{-1}J\end{smallmatrix}\right]\) (\(J\) is a symmetric tridiagonal real matrix, \(R\), \(C\) are positive diagonal matrices, and \(I\) is the identity matrix) and the solutions of the initial value problem \[ {d\over dt} \Biggl[\begin{matrix} u(t)\\ v(t)\end{matrix}\Biggr]= \Biggl[\begin{matrix} 0 & I\\ C^{-1}R & C^{-1}J\end{matrix}\Biggr] \Biggl[\begin{matrix} u(t)\\ v(t)\end{matrix}\Biggr],\quad \Biggl[\begin{matrix} u(0)\\ v(0)\end{matrix}\Biggr]= \Biggl[\begin{matrix} f\\ g\end{matrix}\Biggr] \] are studied. The matrix \(A\) has \(2N\) distinct eigenvalues \(\lambda_j\) different from zero, half of them negative, the other half positive. The properties of eigenvalues and eigenvectors are given. The above initial value problem has a unique solution. Its form is proven.