I am new here and have an interesting question:
Consider the (n x n) symmetric and real Matrix M with
$\sum_j M_{i,j} = 0$
$\sum_i M_{i,j} = 0$
$M_{i,i} > 0$
It seems a matrix of this type is always positive semi-definite. How can this be proven?
If we would also have that $M_{i,j} < 0$ for $i \neq j$, then one could use the Gershgorin circle theorem, together with the fact, that all eigenvalues of $M$ have to be real. However, if $M_{i,j}, i \neq j$ takes positive and negative values, this proof does not work.
I think positive semi-definiteness also holds for the general case of different signs of $M_{i,j}, i \neq j$. I tried many matrices and never found a counter example.
Thanx for your help, Martin