Considering the statement: "Let Q be a symmetric matrix. If there exist positive and negative elements in the diagonal, then Q is indefinite."
Is the contrapositive of the statement: If Q is not indefinite, then all diagonal elements are either not postive or not negative (meaning all diagonals are of one sign).
I thought proving the contrapositive would be easier than the original statement, but I just wanted to make sure the above is actually the contrapositive.
The direct proof is easy. Say without loss of generality that the $q_{1,1}$ is positive and $q_{2,2}$ is negative, then we have that $$ e_1^TQe_1=q_{1,1}>0 $$ and $$ e_2^TQe_2=q_{2,2}<0 $$ so we have that $Q$ represents an indefinite quadratic form.