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...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.展开更多
基金Acknowledgements This work was supported in part by the National Natural Science Foundation of China （Grant Nos. 61301296, 61377006, 61201396） and the National Natural Science Foundation of China-Guangdong Joint Found （No. U1201255）.
文摘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.
