$A$ is a doubly stochastic matrix, how about $A^TA$ I am reading a paper with assumption that $A \in R^{n\times n}$ is a doubly stochastic matrix.
However, the paper says $A^TA$ is symmetric and stochastic. 


*

*Since $A^TA$ is symmetric, if $A^TA$ is stochastic, it must be doubly stochastic.  

*The $(j,j)$ entry of $A^TA$ is exactly $a_{ij}^Ta_{ij}$ with $i=1,\ldots,n$.  
Then is there any other condition I can use to prove this claim is true or false? 

If $A$ is doubly stochastic, then $A^T A$ is doubly stochastic too. Proof is strightforward. Let $B=A^T A$. At first,
$$
b_{ij} = \sum_{k=1}^n (a^T)_{ik} a_{kj} = \sum_{k=1}^n a_{ki} a_{kj}.
$$
Check property of doubly stochastic matrix:
$$
\sum_{j=1}^n b_{ij} = \sum_{j=1}^n \sum_{k=1}^n a_{ki} a_{kj}=
\sum_{k=1}^n \sum_{j=1}^n a_{ki} a_{kj} = \sum_{j=1}^n a_{ki}\sum_{k=1}^n  a_{kj} = \sum_{j=1}^n a_{ki} = 1,
$$
and analogously for $\sum_{i=1}^n b_{ij}$.
