Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

The cone of symmetric positive semidefinite $n\times n$ matrices is the convex hull of rank $1$ matrices. That is, every symmetric positive semidefinite matrix is a convex combination of rank 1 matrices.

Does this property generalize to solutions of linear systems of semidefinite matrices?

Let me be precise. Fix $k$ symmetric $n\times n$ matrices $A_1,\ldots, A_k$. Consider the system of linear equations $\langle A_1,X\rangle=\cdots = \langle A_k, X\rangle = 0$, which you want to solve for a symmetric semidefinite $n\times n$ matrix $X\succeq 0$. Here, the inner product of two matrices is $\langle(a_{ij}),(b_{ij})\rangle=\sum_{i,j = 1}^n a_{ij}b_{ij}$.

The set of solutions $X$ forms a closed convex subcone $C$ of the cone of semidefinite $n\times n$ matrices. Is $C$ the convex hull of its rank 1 matrices? Namely, is every solution a convex combination of rank 1 solutions?

share|improve this question
add comment

2 Answers

up vote 5 down vote accepted

No. Try $n=k=2$ with $A_1 = \pmatrix{1 & 0\cr 0 & -1\cr}$ and $A_2 = \pmatrix{1 & 1\cr 1 & -1\cr}$. The only symmetric matrices $X$ with $(A_1,X) = (A_2,X) = 0$ are multiples of $I$, so there are no rank 1 solutions.

share|improve this answer
add comment

The answer seems obviously "no", since a system of equations need not have any rank one solutions (for example, the solution set can be the line $x I_n,$ where $I_n$ is the identity matrix.)

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

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