In an induction proof of a lemma I would like to prove the following statement.
Let $U$ be a non-empty finite set and let $X_{i,j}$ for all $i\not=j \in U$ be real numbers. Assume for each $j\in U$ we have real numbers $\beta^j_i>0$ for each $j\not= i \in U$ such that $\sum_{i\in U\setminus\{j\}}\beta^j_i X_{i,j}>0$. Can we construct real numbers $\beta_i>0$ such that $ \sum_{j\in U} \sum_{i\in U\setminus \{j\}}\beta_i X_{i,j}>0$.
Obviously $\sum_{j\in U} \sum_{i\in U\setminus \{j\}}\beta_i^j X_{i,j}>0$ and more generally $\sum_{j\in U} a_j\sum_{i\in U\setminus \{j\}}\beta_i^j X_{i,j}>0$ for non-negative number $a_j$, but I need $\beta_i^j$ to be equal for all $j$.
Inspired by above remark I have tried finding positive weights $a_j$ and let $\beta_i = \sum_{j\in U\setminus \{i\}} a_j \beta_i^j$ to no avail.
Defining the matrix $X = \{X_{i,j}\}$ we can possibly rephrase it as some statement about matrices.
This is possibly already known but it is hard to find good terms to search for. If anyone have any feedback I would be very pleased.
