Richard Sinkhorn

Problems involving diagonal products in nonnegative matrices

Nonnegative matrices characterization by diagonal products to imply properties of stochastic matrices

A relationship between arbitrary positive matrices and stochastic matrices

The author (2) has shown that corresponding to each positive square matrix A (i.e. every aij > 0) is a unique doubly stochastic matrix of the form D1AD2, where the Di are diagonal matrices with positive diagonals. This doubly stochastic matrix can be obtained as the limit of the iteration defined by alternately normalizing the rows and columns of A. In this paper, it is shown that with a sacrifice of one diagonal D it is still possible to obtain a stochastic matrix. Of course, it is necessary to modify the iteration somewhat. More precisely, it is shown that corresponding to each positive square matrix A is a unique stochastic matrix of the form DAD where D is a diagonal matrix with a positive diagonal. It is shown further how this stochastic matrix can be obtained as a limit to an iteration on A.

Minimum permanents of doubly stochastic matrices with at least one zero entry

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.

Linear adjugate preservers on the complex matrices

It is shown that if L is a linear map from the set of n × n complex matrices into itself such that L(adj A) = adj L(A) for all A, and if n ≠ 2, then L has one of the forms is a linear combination of maps of the form PA(adj P) and/or QAT (adj Q). The Ps and/or Qs need not be nonsingular when n = 2.

Power symmetric stochastic matrices

Abstract Stochastic matrices A which satisfy the equation AT=Ap are characterized for integral values of p > 1.

A problem related to the van der waerden permanent theorem

Let Kn denote the set of nonnegative n×n matrices whose elements have sum n. It is cOnjectured that if SeKn , then per . wHere r, and c1 are respectively the ith row sum and jth column sum of S, and that equality holds only if S=Jn , the n×n matrix whose entries all equal n−1 . In this paper it is shown that for any SeKn such that per is minimal, per S > 0.

Permanents of special classes of doubly stochastic matrices

We identify the doubly stochastic matrices with at least one zero entry which are closest in the Euclidean norm to Jn , the matrix with each entry equal to 1/n, and we show that at these matrices the permanent function has a relative minimum when restricted to doubly stochastic matrices having zero entries.

Doubly stochastic matrices with dominant p-minors

It has been shown that if A is an n×n doubly stochastic matrix, and if the permanent of A is minimal with respect to all n×n doubly stochastic matrices, then per A(i|j) per A for all i,j=1,…,n . It is conjectured that if A is doubly stochastic and if per A for all i,j=1,…,n, then either A=Jn , the matrix in which every entry is l/n, or, up to permutations of rows and columns, , where Pn is a full cycle permutation matrix. The resolution of this conjecture in the affirmative would settle the famous van der Waerden conjecture concerning permanents of doubly stochastic matrices. Results in this paper confirm the former conjecture under certain restrictions on the matrix A.

On matrices minimizing the permanent on faces of the polyhedron of the doubly stochastic matrices

Let Ωn(Z) denote the set of n× ndoubly stochastic matrices (a ij) with a ij= 0 for (i,j)∊Z. H. Minc conjectured that for a matrix Aminimizing per (A) on Ωn(Z) one has per( A(i\j)) = per (A) if (i j) and per(A(i|j))  per(A) if (i j)eZ. This paper gives counterexamples to both parts of Mincs conjecture. A modified conjecture is proposed.

Concerning the Magnitude of the Entries in a Doubly Stochastic Matrix

We show that for every nxn doubly stochastic matrix A, for every entry aij , where is the Euclidean norm. The cases for which equality holds are discussed in detail.