MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).
show/hide this revision's text 2 added 85 characters in body

This si is motivated by this MO question.

If $A\in{\bf M}_n({\mathbb R})$ is row-stochastic (entrywise non-negative, and $\sum_j a_{ij}=1$ for all $i$), then $M:=A+A^T$ is

  • symmetric,

  • entrywise non-negative.

One finds easily the

  • additional property that $$\sum_{i\in I}\sum_{j\in J}m_{ij}\le|I|+|J|$$ for every index subsets $I$ and $J$,

with

  • equality in the extremal case: $$\sum_{i,j=1}^nm_{ij}=2n.$$

My question is whether all these four properties imply in turns that $M$ has the form $A+A^T$ for some row-stochastic $A$.

Edit. The answer is Yes when $n=2$ (obvious) or $n=3$ (more interesting).
show/hide this revision's text 1

What is $A+A^T$ when $A$ is row-stochastic ?

This si motivated by this MO question.

If $A\in{\bf M}_n({\mathbb R})$ is row-stochastic (entrywise non-negative, and $\sum_j a_{ij}=1$ for all $i$), then $M:=A+A^T$ is

  • symmetric,

  • entrywise non-negative.

One finds easily the

  • additional property that $$\sum_{i\in I}\sum_{j\in J}m_{ij}\le|I|+|J|$$ for every index subsets $I$ and $J$,

with

  • equality in the extremal case: $$\sum_{i,j=1}^nm_{ij}=2n.$$ My question is whether all these four properties imply in turns that $M$ has the form $A+A^T$ for some row-stochastic $A$.