It is known that a 3 by 3 real symmetric matrix $A$ has an eigendecomposition

$$ A = Q E Q^T $$

where $Q$ is an orthogonal matrix and $E$ is a diagonal matrix whose elements, $E_{11}$, $E_{22}$ and $E_{33}$, are the eigenvalues of $A$.

Moreover, if those eigenvalues are non-negative then $A$ is positive-semidefinite.

The question is: **if those eigenvalues are not only non-negative but also verify the triangle inequalities
$$
\begin{aligned}
E_{11} + E_{22} &\geq E_{33}\\
E_{11} + E_{33} &\geq E_{22}\\
E_{22} + E_{33} &\geq E_{11}
\end{aligned}
$$
is there anything special about the structure of $A$ besides the fact that it is positive-semidefinite?**

Can those extra triangle inequalities constraints be written as functions of the elements of $A$, just like the positive-semidefinite constraints can be written down using the Sylvester's criterion?

Can those extra constraints be somehow represented in a semidefinite programming / linear matrix inequalities framework?