The nearest generalized doubly stochastic matrix to a real matrix with the same firstand second moments

Abstract

Abstract. Let T be an arbitrary n × n matrix with real entries. We explicitly find the closest (in Frobenius norm) matrix A to T , where A is n × n with real entries, subject to the condition that A is "generalized doubly stochastic" (i.e. Ae = e and e T A = e T , where e = (1, 1, . . . , 1) T , although A is not necessarily nonnegative) and A has the same first moment as T (i.e. e T 1 Ae 1 = e T 1 T e 1 ). We also explicitly find the closest matrix A to T when A is generalized doubly stochastic has the same first moment as T and the same second moment as T (i.e. e T 1 A 2 e 1 = e T 1 T 2 e 1 ), when such a matrix A exists.Mathematical subject classification: 15A51, 65K05, 90C25.

Hardy-type inequalities for ℓ^p sequences

“…He showed that D = W BW + J n , where W = I n − J n and J n is the n × n matrix with every entry uniformly equal to 1/n. This line of investigation was later pursued by Glunt, Hayden and Reams in [12] and further developed in the more specific case of doubly stochastic matrices (as opposed to generalized doubly stochastic matrices) in [2] by Bai, Chu and Tan in 2007.…”