I have an energy dispersion curve obtained from the eigenvalues of $$E(k) = \text{eig}(T e^{ik} + T^H e^{-ik} + H_0),$$where $H_0$ and $T$ are $N\times N$ square matrices, $T^H$ is the Hermitian transpose $T$, and $k$ is the wave-vector. So, there are just 2 matrices $T$ and $H_0$ that determine the problem.I want to reduce the size of matrices $T$ and $H_0$ in a way that the approximation results in the same eigenvalues for a limited energy range (i.e. $E = [1:3]$). Is it possible to construct new matrices $T_1$ and $H_1$ such that $E_1(k) = \text{eig}(T_1e^{ik} + T_1^H e^{-ik} + H_1)$ be close to $E(k)$ for a limited energy range? How?
Size reduction of matrices in dispersion curve calculation
eigensystem;eigenvalues;basis set