Assume we have the following decomposition: $M=AWA^T$
Where:
$M$ is non-negative symmetric (i.e. $M_{i,j}\ge{0}$ and $M_{i,j}=M_{j,i}$)
$A$ is right stochastic (entries are non-negative and rows sum to 1)
$W$ is non-negative and symmetric
My questions are whether this kind of decomposition has been studied and specifically whether it is unique (excluding permutation and multiplication by a constant).
A symmetric matrix $W = W^T$ can be written as follow $$ W = Q \Lambda Q^T, \qquad\text{(spectral theorem)}$$
where $\Lambda $ is a diagonal matrix and symmetric ($\Lambda = \Lambda^T$), and $Q$ is an orthogonal matrix.
Thus, we can write
$$\begin{align*} M &= AQ\Lambda Q^TA^T \\ &= (AQ)\Lambda (AQ)^T\\ &= C\Lambda C^T \end{align*}$$
where $C=AQ$
If $W$ is real positive-definite, then
$$ W = LL^T \qquad \qquad \text{(using Cholesky decomposition)}$$
Substituting this into the expression of $M$ we get
$$\begin{align*} M &= AWA^T \\ &= ALL^TA^T\\ &= AL(AL)^T \end{align*}$$
So, the decomposition is not unique.