I need to find a symmetric matrix A \begin{pmatrix} a_{11} & a_{12} & a_{13}\\ a_{12} & a_{22} & a_{23}\\ a_{13} & a_{23} & a_{33} \end{pmatrix} such that its' main minors are positive, but not all the elements on diagonal are. I.e. (without loss of generality) the following conditions have to be satisfied:
1) $a_{11}>0$
2) $a_{11}a_{22}-a_{12}^2>0$
3) $\det(A)>0$
4) $a_{33} < 0$
I tried to find it but it seems that there's no such matrix (although I can not prove it). The conditions seem to be connected one to another and probably even contradictive with each other. Maybe, it can not be done with 3 by 3 matrices but can be with ones of higher size?
This cannot be done if your matrix has real entries. By Sylvester's criterion, if all leading principal minors of a real symmetric matrix are positive, the matrix itself must be positive definite and hence its diagonal entries are also positive.
However, it is possible that all leading principal minors of a complex symmetric matrix are positive, while some diagonal entries are negative. Consider for instance $$ A=\pmatrix{1&0&0\\ 0&1&2i\\ 0&2i&-1}. $$