On the Inverse Eigenvalue Problem for Irreducible Doubly Stochastic Matrices of Small Orders

The inverse eigenvalue problem is a classical and difficult problem in matrix theory. In the case of real spectrum, we first present some sufficient conditions of a real r-tuple (for r = 2 ; 3; 4; 5) to be realized by a symmetric stochastic matrix. Part of these conditions is also extended to the complex case in the case of complex spectrum where the realization matrix may not necessarily be symmetry. The main approach throughout the paper in our discussion is the specific construction of realization matrices and the recursion when the targeted r-tuple is updated to a ( r + 1 ) -tuple.