Extremal inverse eigenvalue problem for bordered diagonal matrices

The following inverse eigenvalue problem was introduced and discussed in [J. Peng, X.Y. Hu, L. Zhang, Two inverse eigenvalue problems for a special kind of matrices, Linear Algebra Appl. 416 (2006) 336-347]: to construct a real symmetric bordered diagonal matrix A from the minimal and maximal eigenvalues of all its leading principal submatrices. However, the given formulae in [4, Theorem 1] to compute the matrix A may lead us to a matrix, which does not satisfy the requirements of the problem. In this paper, we rediscuss the problem to give a sufficient condition for the existence of such a matrix and necessary and sufficient conditions for the existence of a nonnegative such a matrix. Results are constructive and generate an algorithmic procedure to construct the matrices.