The matrix M satisifies the following properties:
1. m_{ii}>0
2. m_{pq}m_{qp}=m_{pp}m_{qq}
3. m_{ij}m_{ji}=0 or m_{ii}m_{jj}, i,j not equal to p and q.
Many thanks for your reply.
It holds when M is a 2 by 2 matrix.
I tried using row reduction, most items become same but there still exist some items uncertain.
In fact, I am perplexed by this problem for such a long time and I have tried using induction but failed.