0
$\begingroup$

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.

$\endgroup$
2
$\begingroup$

The set of all matrices with off-diagonal elements negative is a convex set, quite obviously. Intersecting convex sets results in a convex set.

$\endgroup$
  • $\begingroup$ Thank you for your quick response. Do you mean that all the subsets of a convex set are convex? Can you please elaborate? $\endgroup$ – user252783 Jul 24 '16 at 11:14
  • $\begingroup$ @user252783 no, he means that the intersection of two convex sets (here, the set of all positive definite matrices and the set of all matrices having negative off-diagonal elements) is a convex set. $\endgroup$ – user258700 Jul 24 '16 at 11:37
  • $\begingroup$ Okay I am confused, I know that $\endgroup$ – user252783 Jul 25 '16 at 5:56
  • $\begingroup$ I am confused, (1) I know that set of all positive definite matrices take shape of a Cone and is a convex set (2) I read that intersection of two convex sets results in a convex set. My concern is whether the set of _all matrices with off diagonal elements negative and diagonal elements positive which are also positive definite denoted by say set $P_n$_form a convex set or not. Mr. Hagen said set $P_n$ is a convex set. But I did not understand why intersection came into picture. Also said it is obvious that set $P_n$ is convex..I am new in this so appreciate any added description Thank u $\endgroup$ – user252783 Jul 25 '16 at 6:06
  • 1
    $\begingroup$ @user252783 If for each pair $(i,j)$, $C(i,j)$ is a convex set, then the set of matrices $A$ with $A_{ij}\in C(i,j)$ for all $i,j$, is obviously convex. You want to take $C(i,i)=(0,\infty)$ and $C(i,j)=(-\infty,0]$ for $i\ne j$. Thus the set of all matrices with strictly positive diagonal elemnts and with non-positve non-diagonal elements is convex. If we now intersect this convex set with te convex set of all posdef matrices, the result follows. In fact, the posdef matrices can also be written as intersection of many convex sets: For fixed $x$, the set of matrices $A$ with $x^TAx>0$ is convex $\endgroup$ – Hagen von Eitzen Jul 31 '16 at 21:50

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.