TITLE: Structure of a real 3x3 positive-semidefinite matrix whose eigenvalues verify the triangle inequalities
QUESTION [5 upvotes]: 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?

REPLY [8 votes]: 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 
$$A \preceq \frac12(A_{11}+A_{22}+A_{33}) I$$