Before revision:  We refer to the matrices in the two classes as Toeplitz-regularized and  corner-overlapped block-diagonal matrices. The first class generalizes the tridiagonal Toeplitz matrices to matrices with larger bandwidths where boundary entries are modified. For the second class, we decompose the problem into a lower two-block matrix problem where one of the blocks is a quadratic eigenvalue problem (QEVP) .
After revision:  We refer to the matrices in the two classes as Toeplitz-regularized and corner-overlapped block-diagonal while the IGA matrices are almost Toeplitz-plus-Hankel. In fact, IGA with a correction that results in Toeplitz-plus-Hankel matrices gives a better numerical method. In this paper, we focus on finding the analytical eigenpairs to the GEVPs while developing better numerical methods is our motivation .