# Eigenvalues of conditionally positive definite Hermitian matrices

I would like to know if there are any eigenvalue characterizations of a Hermitian $n^2 \times n^2$ matrix $A$ that satisfies

$$(\vec X)^\prime A \vec X \gt 0$$

for all traceless nonzero Hermitian $n\times n$ matrices $X$.

Here $\vec X$ refers to the column vector formed by stacking the columns of $X$ from left to right, and prime denotes conjugate transpose.