QCLP constraint positive semidefinite

Question

I have quadratically constrained linear program where one objective is of the form $x_1.x_2 \leq x_1k_1 + k_2$, $k_1, k_2$ are constants. I know that QCLP is NP-Hard in general but can be solved in PTIME if the matrix for the quadratic constraint is positive semidefinite. For my problem, the matrix is $\begin{bmatrix} 0 & 1/2 \\ 1/2 & 0 \end{bmatrix}$. By the design of my problem, $x, y > 0$. Is positive semidefinite condition $x^T M x \geq 0$ satisfied for all $x$ or only over the the constraints that vector $x > 0$ (in which case, the matrix is positive semidefinite)?

Answer 1

According to Matrix theory (Horn and Johnson), if the diagonals of a matrix are all 0, the matrix can only be indefinite. You may also check with diagonally dominant condition, which (in my impression) is a sufficient condition for positive definiteness.

In your case, the matrix [0, 1/2;1/2 0] has eigenvalues -1/2 and 1/2. So it is indefinite.