Is there a way to take a general $N \times N$ matrix $Y$, and turn it into a double stochastic matrix $X$ analytically?
- With any iterative algorithms?
Ans: The Sinkhorn algorithm does that, but iteratively. One needs to alternatively normalize raws and columns for several iterations.
Richard Sinkhorn: A relationship between arbitrary positive matrices and doubly stochastic matrices, The Annals of Mathematical statistics, 35 (1964), 876–879.
Theorem. If $A$ is a square matrix with strictly positive entries then there are a unique doubly stochastic matrix $T_A$ and diagonal matrices $D_1$, $D_2$ such that $T_A=D_1AD_2$. The matrices $D_1$ and $D_2$ are themselves unique up to a scalar factor.
See How to convert any matrix to doubly stochastic matrix?
- Without any iterative algorithms?
