Inverse eigenvalue problems for composite banded tridiagonal matrices: symmetric and nonsymmetric cases

We study an inverse eigenvalue problem associated with a special class of structured matrices in both symmetric and nonsymmetric forms. In the symmetric case, the problem involves two sets of eigenpairs corresponding to the original matrix and its leading principal submatrix of order $n-1$. For the nonsymmetric case, the minimum and maximum eigenvalues of each leading principal submatrix, together with an eigenpair associated with the maximum eigenvalue of the original matrix, are prescribed. By establishing recurrence relations between successive principal submatrices, we derive the necessary and sufficient conditions for the solvability of the problems. Moreover, explicit reconstruction algorithms are developed, and several numerical results are presented to illustrate the effectiveness of the proposed methods.