$A$ 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? 

I'm not sure I follow what you're trying to prove from which assumptions; I'll assume that the aim is to show that $A^TA$ is doubly stochastic if $A$ is doubly stochastic. You've already shown that since $A^TA$ is symmetric it suffices to show that it is right stochastic. Being right (left) stochastic means having the vector $e$ with all entries $1$ as a right (left) eigenvector with eigenvalue $1$. We have
$$A^TAe=A^Te=(e^TA)^T=(e^T)^T=e\;,$$
where the first equality holds because $A$ is right stochastic and the third one holds because $A$ is left stochastic.
