Is it somehow possible to obtain the eigenvalues and eigenvectors of very large matrices similar to the one below analytically ? $$X= {\begin{bmatrix} 6a & -3b & 0 & 0 & -3b & 0 & 0 & 0 & 0 & 0\\ -\sqrt{3}b & 5a & -2b & 0 & -b & -\sqrt{2}b & 0 & 0 & 0 & 0\\ 0 & -2b & 8a & -\sqrt{3}b & 0 & -\sqrt{2}b & -b & 0 & 0 & 0\\ 0 & 0 & -\sqrt{3}b & 15a & 0 & 0 & -\sqrt{3}b & 0 & 0 & 0\\ -\sqrt{3}b & -b & 0 & 0 & 5a & -\sqrt{2}b & 0 & -2b & 0 & 0\\ 0 & -\sqrt{2}b & -\sqrt{2}b & 0 & -\sqrt{2}b & 5a & -\sqrt{2}b & -\sqrt{2}b & -\sqrt{2}b & 0\\ 0 & 0 & -b & -\sqrt{3}b & 0 & -\sqrt{2}b & 9a & 0 & -2b & 0\\ 0 & 0 & 0 & 0 & -2b & -\sqrt{2}b & 0 & 8a & -b & -\sqrt{3}b\\ 0 & 0 & 0 & 0 & 0 & -\sqrt{2}b & -2b & -b & 9a & -\sqrt{3}b\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -\sqrt{3}b & -\sqrt{3}b & 15a \end{bmatrix}} $$
Is it really possible to find exact solutions of such large matrices, if not how do I know I can't ?
Note: $a$, $b$ are constants