Counting extreme U₁ matrices and characterizing quadratic doubly stochastic operators

Abstract

The U₁ matrix and extreme U₁ matrix were successfully used to study quadratic doubly stochastic operators by R. Ganikhodzhaev and F. Shahidi [Linear Algebra Appl., 2010, 432: 24–35], where a necessary condition for a U₁ matrix to be extreme was given. S. Yang and C. Xu [Linear Algebra Appl., 2013, 438: 3905–3912] gave a necessary and sufficient condition for a symmetric nonnegative matrix to be an extreme U₁ matrix and investigated the structure of extreme U₁ matrices. In this paper, we count the number of the permutation equivalence classes of the n × n extreme U₁ matrices and characterize the structure of the quadratic stochastic operators and the quadratic doubly stochastic operators.

Keywords

Extreme U₁ matrix / quadratic doubly stochastic operator / majorized / permutation similar / irreducible matrix

References

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[7] Yang S, Xu C. On extreme U₁matrices. Linear Algebra Appl, 2013, 438: 3905–3912

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2014 Higher Education Press and Springer-Verlag Berlin Heidelberg