Two inverse eigenvalue problems for a special kind of matrices

In this paper, a special kind of matrices which are symmetric, all elements are equal to zero except for the first row, the first column and the diagonal elements, and the elements of the first row are positive except for the first one are considered. Two inverse problems are discussed. One is to construct one of this kind of matrices by the minimal and maximal eigenvalues of its all leading principal submatrices. The other is to construct one of this kind of matrix by one of its eigenpair and eigenvalues of its all leading principal submatrices. The necessary and sufficient conditions for the solvability of the two problems are derived. Furthermore, corresponding numerical algorithms and some examples are given.