Minimum permanents of doubly stochastic matrices with at least one zero entry†
Paul J. Knopp, Richard Sinkhorn · Linear and Multilinear Algebra · 1982
It is shown that the minimum value of the permanent on the n× ndoubly stochastic matrices which contain at least one zero entry is achieved at those matrices nearest to Jn in Euclidean norm, where Jn is the n× nmatrix each of whose entries is n-1 . In case n ≠ 3 the minimum permanent is achieved only at those matrices nearest Jn ; for n= 3 it is achieved at other matrices containing one or more zero entries as well.