Let $A$ denote a symmetric matrix of non-negative entries, whose rows (and columns) sum up to the all-positive vector $d := A\mathbf{1}$ with $\mathbf{1}$ the all-one-vector. Denote $D := diag(d)$ by a capital letter, similar for any other vector.
Let $b$ denote any all-positive vector.


Assume there exists a positive diagonal matrix $S$ such that $A^\*:=SAS$ provides $b$ as its row (and column) sums, that is $A^\*\mathbf{1} = b$.


It is well-known that **iterated proportional fitting** provides a sequence $(A=:\tilde A\_{(0)}, \tilde A\_{(1)}, \tilde A\_{(2)}, \ldots)$ that converges to $A^\*$. It *alternately* scales rows appropriately, then columns, then rows, and so forth. This implies that no $\tilde A\_{(k)}$ for $k>0$ is symmetric.


However, I am interested in the following **symmetrization**:


1. Set $A\_{(0)} := A$ and $k:=0$
2. Set $d\_{(k)} := A\_{(k)} \mathbf{1}$
3. Set $A\_{(k+1)} := B^{1/2}D\_{(k)}^{-1/2} \cdot A\_{(k)} \cdot D\_{(k)}^{-1/2}B^{1/2}$
4. Increase $k \gets k+1$ and continue at Step 2


This provides a sequence of *symmetric* matrices $(A=:\tilde A\_{(0)}, \tilde A\_{(1)}, \tilde A\_{(2)}, \ldots)$.
It is known that it converges to $A^\*$ in the case that $b = \mathbf{1}$, thus, if the limit is doubly stochastic.


But does this also hold for $b \ne \mathbf{1}$?


I strongly believe this, but I cannot find any results on this, and the proof of the case $b = \mathbf{1}$ seems not to generalize.



Convergence of Symmetric Iterative Proportional Fitting
Now, one year later, I found a recent publication by S. Kurras that answers my above question in all aspects. See here for the paper plus its supplement:
[Proceedings of AISTATS 2015](http://www.jmlr.org/proceedings/papers/v38/)


Indeed he proves that SIPF converges to the same limit as IPF, for any all-positive vector $b$.


This further implies that the limit matrix $A^\*$ is precisely the Bregman projection of $A$ onto the set $\mathcal{S}(b,A)$ defined by all symmetric non-negative matrices of row/column sums $b$, where the projection is with respect to relative entropy error (generalized Kullback-Leibler divergence)
$$D(X\|Y) := \sum\_{i,j} x\_{ij} \log(x\_{ij}/y\_{ij}) - x\_{ij} + y\_{ij}.$$


Thus, it holds that both SIPF and IPF iteratively approximate the following optimum/projection:


$$A^\* \ = \ \text{arg}\,\text{min}\_{X \in S(b,A)} \ D(X\|A)$$


(whenever existing, i.e., whenever $\mathcal{S}(b,A) \ne \emptyset$).


