Notes on permanents of doubly stochastic matrices
Derek K Chang · Linear and Multilinear Algebra · 1983
Using a simple probabilistic interpretation of the permanents of doubly stochastic matrices, we obtain in this paper some results which extend the work in [1]. An upper bound is obtained for the permanent on the doubly stochastic matrices A=(aij) with all of whose entries satisfying the condition . It is proved that for doubly stochastic matrices in the complement of a sufficiently large neighborhood of Jn , a conjecture of E. T. H. Wang holds, which asserts that per be the set of all n×n doubly stochastic matrices with at least one entry equal a. A necessary and sufficient condition on a is given such that the permanent function restricted to Ωn(a) achieves a local minimum at the matrices which are closest to Jn in Euclidean norm in Ωn(a).