This question is related to my previous question Set of all positive definite matrices with off diagonal elements negative
I know that the shape of "the space of the set of all positive definite matrices (PDM)" is a convex cone.
What will be geometrically the shape of "the space of the set of all positive definite matrices of which off diagonal elements are negative (PDM_n)?.
Such positive matrices with off diagonal elements negative (PDM_n) exists. For instance, the case with n=2, the symmetric mat $A=[a_1, a_2;a_2, a_3]$ for being a PDM we have the condition, $x'Ax>0$, $a_2\neq 0$, $x\neq 0$ where $x=[x_1,x_2]'$. Also we have $a_1>0$ and $a_3>0$ as they are principal minors.
I see that $x_1^2a_1+2x_1x_2a_2+x_2^2a_3$ can be positive if 1) $x_1$ and $a_2$ negative and $x_2$ positive 2) $x_2$ and $a_2$ negative and $x_1$ positive, 3) $x_1>0$, $x_2>0$ and $a_2<0$. Also note that 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 $2x_1x_2a_2$.
But with above analysis I am not able to picturize the shape of "the space of PDM_n matrices". Especially for higher dimension such as n=5, what would be the shape of set of all matrices in $\mathbb{R}^{5\times 5}$ with off diagonal elements negative. I would like to know whether the space of PDM_n is convex or not. Thank you.
The set of all matrices with off-diagonal elements negative is a convex set, quite obviously. Intersecting convex sets results in a convex set.