Inverse eigenvalue problem of nonnegative matrices via unit lower triangular matrices (Part I)

This paper uses unit lower triangular matrices to solve the nonnegative inverse eigenvalue problem  for various sets of real  numbers. This problem  has remained unsolved for many years for $n \geq 5.$  The inverse of the unit lower triangular matrices can be easily calculated and the matrix similarities are also helpful to be able to solve this important problem to a considerable extent. It is assumed that in the given set of eigenvalues, the number of positive eigenvalues is less than or equal to the number of nonpositive  eigenvalues to find a nonnegative matrix such that the given set is its spectrum.