The Non-negative Inverse Eigenvalue Problem for Tridiagonal Matrix, Circulant Matrix and Symmetric Matrix

The Non-negative Inverse Eigenvalue Problem (NIEP) is a sub-problem extracted from inverse eigenvalue problem with a long history from 1930s determining the sufficient and necessary conditions in order that, σ={^λ₁,...,^λ_n}to be the spectrum of an entry wise non-negative n x n matrix. There had been many excellent researchers and scholars who contributed to discover many practical theories. The following of this paper would be then separated into two parts to further analyze the NIEP. In the first section of the paper, some important preexisting conclusions would be demonstrated and the groups' understanding of these indispensable theories would be expressed. In the second section, three special and wildly used matrix, including tridiagongal matrix, circulant matrix, and symmetric matrix would be considered. The solution of the NIEP of these matrices done by the group would be expressed.