Counting extreme \(U_1\) matrices and characterizing quadratic doubly stochastic operators
The \(U_1\) matrix and extreme \(U_1\) matrix were successfully used to study quadratic doubly stochastic operators by \textit{R. Ganikhodzhaev} and \textit{F. Shahidi} [Linear Algebra Appl. 432, No. 1, 24--35 (2010; Zbl 1186.15025)], where a necessary condition for a \(U_1\) matrix to be extreme was given. \textit{S. Yang} and \textit{C. Xu} [Linear Algebra Appl. 438, No. 10, 3905--3912 (2013; Zbl 1305.15025)] gave a necessary and sufficient condition for a symmetric nonnegative matrix to be an extreme \(U_1\) matrix and investigated the structure of extreme \(U_1\) matrices. In this paper, the authors count the number of the permutation equivalence classes of the \(n\times n\) extreme \(U_1\) matrices and characterize the structure of the quadratic stochastic operators and the quadratic doubly stochastic operators.
scientific article; zbMATH DE number 6714854
scientific article