On a conjecture of R. A. Brualdi
Let \(n>2\). Let \(X_ n=(x_{ij})\) denote the \(n\times n\) matrix that has \(x_{ij}=(n-1)/n(n-2)\) if \(i\neq j\), \(x_{ij}=0\) if \(i=j\), \(i,j=1,...,n- 1\); \(x_{ij}=1/n\) if either i or j equals n. The author refutes a conjecture of \textit{R. A. Brualdi} [Linear Multilinear Algebra 17, 5-18 (1985; Zbl 0564.15010)] that \(X_ n\) has the unique minimal permanent among all \(n\times n\) doubly stochastic matrices with zeros in positions (i,i), \(i=1,...,n-1\), for \(n\geq 5\). The minimum permanent is determined for a subset of such doubly stochastic matrices.
scientific article; zbMATH DE number 4014912