Matrix Determinant Inequality And The Inverse Eigenvalue Problem

This paper consists of two parts. In the first part, we shall prove several inequalities involving symmetric positive semidefinite, general M-matrices and inverse M-matrice which are generalization of the classical Oppenheim's Inequality for symmetric positive semidefinite matrices. Those main proofs of this part are based on nonnegative. And in the second part, we consider the following four inverse eigenproblems: the reconstruction of normal five-diagonal matrix by two or three ordered eigenvalues and corresponding eigenvectors, and the reconstruction of real symmetric tridiagonal matrix and irreducible real symmetric tridiagonal matrix by three eigenvalues and corresponding eigenvectors, some sufficient conditions for existence of unique solution to the problems are given here, and some necessary and sufficient conditions for the existence of both unique solution and solution (not unique) to the latter are also given. Meanwhile, explicit expressions of each unique solution are given. Finally, numerical experiments are given.