Counting extreme U1 matrices and characterizing quadratic doubly stochastic operators  

作　　者：Quanbing ZHANG Shangjun YANG 

机构地区：[1]Key Laboratory of Intelligent Computing & Signal Processing, Ministry of Education,Anhui University, Hefei 230039, China

出　　处：《Frontiers of Mathematics in China》2016年第3期647-659,共13页中国高等学校学术文摘·数学（英文）

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）.

摘　　要：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.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.

关 键 词：Extreme U1 matrix quadratic doubly stochastic operator majorized permutation similar irreducible matrix 

分 类 号：O177.91[理学—数学] O151.21[理学—基础数学]