Please, help me to find the set of all positive definite matrices (PDM) of which off diagonal elements are negative.
Considering the case with n=2, the symmetric mat $A=[a_1, a_2;a_2, a_3]$ needs to be a PD. i.e. $x'Ax>0$, $a_2\neq 0$, $x\neq 0$ where $x=[x_1,x_2]'$.
Always we have $a_1>0$ and $a_3>0$ as they are principal minors.
I guess such matrices exists as $x_1^2a_1+2x_1x_2a_2+x_2^2a_3$ can be positive with either of $x_1$ or $x_2$ negative and $a_2$ negative.
Also, in case which $x_1>0$, $x_2>0$ and $a_2<0$, the condition $x'Ax>0$ is met if the absolute value of $x_1^2a_1+x_2^2a_3$ is greater than absolute value of $2x_1x_2a_2$.
How can I genearlize this to denote and derive set of all possible positive definite matrices? I mean can you give some examples of A which is positive definite with offdiagonal elements negative? or suggest method to get them? please?