Concerning hypernormal stochastic matrices (Q923669)

An \(n\times n\) matrix A is hypernormal with respect to a set S of \(n\times n\) matrices if \(AXA^ T=A^ TXA\) for all \(X\in S\). The author established some characterizations concerning hypernormal stochastic matrices: If A is a hypernormal stochastic matrix with respect to generalized stochastic matrices (i.e. real matrices with unit row sums), then either A is symmetric or \(a_{ij}+a_{ji}=1/(2n)\). If A is nonnegative hypernormal with respect to doubly generalized matrices, then either A is symmetric or A is a positive multiple of a doubly stochastic matrix. There are some other properties as well.