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In this note, we present an algorithm that yields many new methods for constructing doubly stochastic and symmetric doubly stochastic matrices for the inverse eigenvalue problem. In addition, we introduce new open problems in this area that lay the ground for future work.(possibly complex) to be the spectrum of an n × n nonnegative matrix A. Although this inverse eigenvalue problem has attracted a considerable amount of interest, for n > 3 it is still unsolved except in restricted cases. Generally, we have two cases that result in three problems.• When λ = (λ 1 , ..., λ n ) is complex, little is known. The case n ≤ 3 was completely solved in [17] and the solution of the 4 × 4 trace-zero nonnegative inverse eigenvalue problem was given in [23].• When the n-tuples (λ 1 , . . . , λ n ) are real, then we have the following two problems: 1) The real nonnegative inverse eigenvalue problem (RNIEP) asks which sets of n real numbers occur as the spectrum of an n × n nonnegative matrix A.2) The symmetric nonnegative inverse eigenvalue problem (SNIEP) asks which sets of n real numbers can occur as the spectrum of an n × n symmetric nonnegative matrix A.Each problem remains open. Many partial results for the three problems are known see the recent book [10] and the references therein, for the collection of all known results concerning these problems. Various people raised the question whether the (RNIEP) and (SNIEP) are generally equivalent. In the low dimension n ≤ 4, the two problems are actually equivalent (see [9]). However, for n > 4, the paper [14] showed that the two problems are generally different. In addition, the paper [9] gives a construction for data λ = (λ 1 , ..., λ 5 ) which is a solution for the (RNIEP) and there is no symmetric nonnegative 5 × 5 matrix with spectrum λ.Another object of study in this area that has a big interest is the inverse eigenvalue problem for nonnegative matrices with extra properties. For example, we can consider the same problems for doubly stochastic matrices. So that we have the following problems.Problem 1.1. The doubly stochastic inverse eigenvalue problem denoted by (DIEP), is the problem of determining the necessary and sufficient conditions for a complex n-tuples to be the spectrum of an n × n doubly stochastic matrix. Equivalently, this problem can also be characterized as the problem of finding the region Θ n of C n such that any point in Θ n is the spectrum of an n × n doubly stochastic matrix. Now, when the n-tuples (λ 1 , . . . , λ n ) are all real, then we have the following two problems:

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In this paper, we study the inverse eigenvalue problem for n × n symmetric doubly stochastic matrices. The spectra of all indecomposable imprimitive symmetric doubly stochastic matrices are characterized. Then we obtain new sufficient conditions for a real n-tuple to be the spectrum of an n × n symmetric doubly stochastic matrix of zero trace. Also, we prove that the set where the decreasingly ordered spectra of all n × n symmetric doubly stochastic matrices lie is not convex. As a consequence, we prove that the set where the decreasingly ordered spectra of all n × n non-negative matrices lie is not convex.