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?



Structure of a real 3x3 positive-semidefinite matrix whose eigenvalues verify the triangle inequalities
Condition 
$$E\_{11}+E\_{22}\ge E\_{33}$$
is equivalent to 
$$E\_{11}+E\_{22}+E\_{33}\ge 2 E\_{33}$$
or
$$E\_{33} \le \frac12 tr\ A$$
since the sum of eigenvalues is equal to the trace. Combining with the other two conditions gives
$$\lambda\_{max}(A) \le \frac12 tr\ A$$ 
which is semidefinite representable as 
$$A \preceq \frac12(A\_{11}+A\_{22}+A\_{33}) I$$


