How do you minimize a positive semidefinite matrix with trace constraint?

Question

Let $X$ be the matrix that you need to optimize. The problem is $$\min \qquad \frac{1}{a^T X a}$$ $$\text{subject to} \qquad trace(X) = 1$$ $$\qquad X \succeq 0$$

Here $a \in \mathbb{R}_+^2$. In my opinion, since it is like in the form of $\frac{1}{x}$ (and hence non convex?) in the positive plane for 1 dimensional case, the optimal value should be as big x as possible (1 in the case of 1 dimension).

Answer 1

$$\min \qquad \frac{1}{a^{\top} X a}$$ $$\text{subject to} \qquad \mathrm{trace}(X) = 1$$ $$\qquad \ \ \ \ \ X \succeq 0$$ The optimization problem can also be posed as follows, $$-\min \qquad \mathrm{trace}(XA)$$ $$\text{subject to} \qquad \mathrm{trace}(X) = 1$$ $$\qquad \quad X \succeq 0$$ where $A=-aa^{\top}$. This is an SDP. If I am not wrong then this is the SDP relaxation of the MAX-CUT problem.