# Converting a Doubly Stochastic Matrix into a Double Stochastic Positive Operator

I am currently studying a paper on operator scaling. Let $$M(N)$$ be the space of all complex $$N \times N$$ matrices. My goal is to understand the analogy between non-negative matrices $$A \in M(N)$$, and positive operators $$T: M(N) \to M(N)$$, i.e. linear maps $$T$$ such that for any positive-semidefinite matrix $$X$$, $$T(X)$$ is positive-semidefinite.

Given a non-negative matrix $$A$$, we can define a row rescaled matrix $$A'$$ by setting

$$A_{ij}' = \frac{A_{ij}}{\sum_{j'} A_{ij'}}$$

Then each row of $$A$$ sums up to one. Similarily, we can define a column rescaling of $$A$$. The goal here is to construct matrices which are close to being doubly stochastic, matrices whose rows and columns all sum to one.

The analogous operation for a positive operator $$T: M(N) \to M(N)$$ is to consider the `row rescaled' positive operator $$T': M(N) \to M(N)$$ by setting $$T'(X) = T(I)^{-1/2} \cdot T(X) \cdot T(I)^{-1/2}.$$ Then $$T'(I) = I$$, which is analogous to the fact that the rows of a rescaled matrix sum up to one. The situation of rescaling positive operators seems more complicated than that of rescaling non-negative matrices, so my question is whether the latter is a special case of the former. More precisely, is there a map which associates with each non-negative $$N \times N$$ matrix $$A$$ a positive operator $$T_A: M(N) \to M(N)$$, such that for any matrix $$A$$, the row-rescaling $$A'$$ is associated with the rescaling of $$T_A$$, i.e. $$T_{A'} = T_A'$$.

We can associate to any non-negative matrix $$B\in M(N)$$ a positive map $$T_B:M(N)\rightarrow M(N)$$ in the following way.

Let $$e_1,\ldots,e_N$$ the canonical basis of $$\mathbb{C}^N$$ and define $$T_B(X)=\sum_{i=1}^N\sum_{j=1}^NB_{ij}tr(Xe_je_j^t)e_ie_i^t,$$ where $$tr(\cdot)$$ stands for the trace.

Next, notice that if $$X$$ is a positive semidefinite Hermitian matrix then $$tr(Xe_je_j^t)\geq 0$$, hence $$\sum_{j=1}^NB_{ij}tr(Xe_je_j^t)\geq 0$$.

Thus $$T_B(X)$$ is a non-negative diagonal matrix, whenever $$X$$ is a positive semidefinite Hermitian matrix. So $$T$$ is a positive map.

Notice that $$T_B(Id)$$ is a diagonal matrix such that $$T_B(Id)_{ii}=\sum_{j=1}^NB_{ij}>0$$.

So $$T_B(Id)^{-\frac{1}{2}}T_B(X)T_B(Id)^{-\frac{1}{2}}=T_B(Id)^{-\frac{1}{2}}\left(\sum_{i=1}^N\sum_{j=1}^NB_{ij}tr(Xe_je_j^t)e_ie_i^t\right)T_B(Id)^{-\frac{1}{2}}$$

$$=\sum_{i=1}^N\sum_{j=1}^NB_{ij}tr(Xe_je_j^t)T_B(Id)^{-\frac{1}{2}}e_ie_i^tT_B(Id)^{-\frac{1}{2}}$$

$$=\sum_{i=1}^N\sum_{j=1}^N\frac{B_{ij}}{\sum_{j=1}^NB_{ij}}tr(Xe_je_j^t)e_ie_i^t=T_{B'}(X),$$

where $$B'$$ is obtained from $$B$$ by scaling its rows $$(B'$$ is row stochastic$$)$$.

You asked whether it is possible to associate a positive map to a non-negative matrix such that the scaling algorithm for positive maps and non-negative matrices would be "compatible" and we saw above that the answer is yes.

I would like to point out that the opposite association (from a positive map, we obtain a non-negative matrix) is also very useful idea in studying the scaling algorithm and extensions of Sinkhorn-Knopp theorem to positive maps.

One natural way to connect a positive map $$T:M(N)\rightarrow M(N)$$ to a non-negative matrix is by choosing any pair of orthonormal bases of $$\mathbb{C}^N$$ - $$v_1,\ldots,v_N$$ and $$w_1,\ldots,w_N$$ - and defining $$A\in M(N)$$ such that $$A_{ij}=tr(v_iv_i^*T(w_jw_j^*)).$$

Notice that $$A_{ij}=tr(v_iv_i^*T(w_jw_j^*))\geq 0$$ and if $$T(Id)=Id$$ then $$A$$ is row stochastic, because $$\sum_{j=1}^NA_{ij}=\sum_{j=1}^Ntr(v_iv_i^*T(w_jw_j^*))=tr(v_iv_i^*T(Id))=tr(v_iv_i^*)=1.$$

(If you want $$A$$ to be column stochastic, you must ask $$T^*(Id)=Id$$, which is equivalent to the trace preserving property for $$T$$).

Many important ideas used in Sinkhorn-Knopp theorem for matrices, such as the concepts of support and total support, can now be adapted to positive maps like this:

The positive map $$T:M(N)\rightarrow M(N)$$ has support (or total support) if for any pair of orthonormal bases of $$\mathbb{C}^N$$ - $$v_1,\ldots,v_N$$ and $$w_1,\ldots,w_N$$ - the matrix $$A_{ij}=tr(v_iv_i^*T(w_jw_j^*))$$ has support (or total support).

For example, this connection between positive maps and non-negative matrices was used in this paper to extend Sinkhorn-Knopp theorem to rectangular positive maps $$T:M(N)\rightarrow M(K)$$ $$(N\neq K)$$.