2.  The Sinkhorn-Knopp Algorithm and Fixed Point Problem  Solutions for 2 × 2 and special n × n cases  Circulant matrices for 3 × 3 case  Ongoing work

3.  In the 1960s Sinkhorn and Knopp developed an algorithm which transforms any positive matrix A into a doubly stochastic matrix by pre- and post-multiplication of diagonal matrices where is a solution to where (-1) is the entry-wise inverse.

4. Solutions to for

10. or any other multiple thereof. Solutions to for

11.  All solutions to result in matrices with row and column sums of 1.  Exactly one solution is guaranteed to result in a doubly stochastic matrix if A is all positive. It is the unique solution found by the Sinkhorn- Knopp Fixed Point Algorithm

12.  The general formula for the 2 × 2 case is simple.  The general formula for the 3 × 3 case is far more complicated. Its only real use thus far is to verify that there are at most 6 solutions, which we had already predicted by numerically finding solutions.  We have guesses for how many solutions there are in the general n × n case.

13.  For all non-zero diagonal matrices, every non- zero vector is a solution.  For constant matrices, the only solution is the vector of all 1’s.  Any nonzero multiple of a solution is also a solution—this is especially important.

14. If, and then.

15.  For.  The matrix formed by swapping any two rows of A has the same solutions.  The matrix formed by swapping two columns of A has the same solutions, but with corresponding elements swapped. That is, if columns i and j are swapped in A, then the solution is the original solution, but with elements i and j swapped.

16.  For 3 x 3 and larger matrices, the general case is too complicated.  We considered special cases:  Already mentioned diagonal and constant matrices  Upper and lower triangular  Patterned matrices, including circulant matrices

17.  We have shown that any eigenvector for any non-zero n × n circulant matrix is a solution:  This include the vector of all 1’s, the only solution that results in a doubly stochastic matrix.

18.  Observations:  since.  If, then.

19.  Eigenvectors: if, then

20.  There are other solutions as well. We consider the 3 × 3 case which has three other (non-e.vector) solutions:

21.  There are other solutions as well. We consider the 3 × 3 case which has three other (non-e.vector) solutions:

22.  The columns of those solutions form another circulant matrix, which we label A 1 :  The solutions for this matrix are similar to those for the original circulant matrix, A, which we now label as A 0.

23.  For the first element of the first solution vector is

24.  With similar results for the second and third elements, the first solution vector is

25. after factoring out the common term.

26.  With similar results for the other two solutions, we find all three solutions and form a new (and again circulant) matrix

27.  For the first element of the first solution vector is

28.  With similar results for the second and third elements, the first solution is

29. after factoring out the common term.

30.  With similar results for the other two solutions, we find all three solutions and form a new (yet again circulant) matrix which is, of course, the original matrix A.

32.  Solutions for n × n cases:  Circulant  Upper/lower triangular  Other patterned matrices  Numerical solutions for general case  Maximum number of solutions  How to characterize solutions: do doubly stochastic solutions minimize some sort of energy function for a given matrix, while the non-doubly stochastic solutions maximize the energy function?

33. Questions?