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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. $T_{A'} = T_A'$.

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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$)$.

This answers your question.

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 $T:M(N)\rightarrow M(K)$ $(N\neq K)$.

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