Frontiers of Mathematics in China >
Counting extreme U1 matrices and characterizing quadratic doubly stochastic operators
Received date: 04 Mar 2015
Accepted date: 29 Feb 2016
Published date: 17 May 2016
Copyright
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.
Quanbing ZHANG , Shangjun YANG . Counting extreme U1 matrices and characterizing quadratic doubly stochastic operators[J]. Frontiers of Mathematics in China, 2016 , 11(3) : 647 -659 . DOI: 10.1007/s11464-016-0533-z
1 |
Berman A, Johnson C R. Nonnegative Matrices in the Mathematical Sciences. New York: Academic Press, 1979
|
2 |
Birkhoff G. Three observations on linear algebra. Rev Univ Nac Tucuman Ser A, 1946, 5: 147–151
|
3 |
Ganikhodzhaev R, Shahidi F. Doubly stochastic quadratic operators and Birkhoff’s problem. Linear Algebra Appl, 2010, 432: 24–35
|
4 |
Horn R A, Johnson C R. Matrix Analysis. Cambridge: Cambridge University Press, 1985
|
5 |
Otter R. The number of trees. Ann of Math, 1948, 49: 583–599
|
6 |
Shahidi F. On dissipative quadratic stochastic operators. Appl Math Inf Sci, 2008, 2: 211–223
|
7 |
Yang S, Xu C. On extreme U1matrices. Linear Algebra Appl, 2013, 438: 3905–3912
|
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