On a spectral property of doubly stochastic matrices and its application to their inverse eigenvalue problem

Generalization of some results concerning eigenvalues of a certain class of matrices and some applications

“…On the other hand, if A and B are of trace-zero or α = 0 and T 2 is of trace-zero, then D is of trace-zero. As mentioned earlier for the nonnegative inverse eigenvalue problem and in a similar fashion, the preceding two theorems have obvious applications in the study of the inverse eigenvalue problem for doubly stochastic matrices (see [17]). …”

“…We conclude this section by noting that in the preceding two theorems if T 1 and T 2 are symmetric then D is necessarily symmetric. Again the preceding two results are useful in the study of the inverse eigenvalue problem for symmetric doubly stochastic matrices (see the techniques used in [17]). …”

On Spectral Properties of Doubly Stochastic Matrices

“…For more details, we refer to [11][12][13]. [14] showed that the eigenvector has the form x = 1 √ n (1, 1, 1, ..., 1) t corresponding to eigenvaluê λ for x ∈ R n , where R denotes the real line. The spectrum of a doubly stochastic matrix is bounded by 1, that is λ i ≤ 1 for all i.…”

“…The spectrum of a doubly stochastic matrix is bounded by 1, that is λ i ≤ 1 for all i. Three important eigenvalue problems for doubly stochastic matrices are considered in [14] whenever there is a possibility that eigenvalues can be placed in complex plane, denoted by C. The first problem deals with necessary and sufficient conditions for n−tuples to be the spectrum of a given doubly stochastic matrix. The second problem is about the fact that which real numbers acts as the spectrum of the doubly stochastic matrix.…”

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

“…Sufficient conditions for the existence of an entrywise positive matrix with ( ) = Λ have been investigated by many authors [1][2][3][4][5][6][7][8][9][10][11][12][13][14]. The case = 2 is trivial.…”