it’s well known that a matrix $A∈ M_n (R)$ has the following decomposition: $$A=\frac12(A+A^T)+\frac12(A-A^T)=S+K$$ with $S=\frac12(A+A^T)$ and $K=\frac12(A-A^T)$ where $S$ is the symmetric component of $A$ and $K$ is the skew-symmetric component of $A$.
Now, my problem is: given a symmetric matrix $S$, is it possible to find its skew symmetric complement matrix $K$ so that the resulting matrix $A=S+K$ is normal or at least belong to a sub-class of normal operators (quasi normal, subnormal, hyponormal or even paranormal).
$\def\Com{\operatorname{Com}}$ Recall that a matrix (or the corresponding linear operator) is
Normal if its self-commutator $\Com(A)=AA^T-A^T A=0$ or if $\Com(S,K)=SK-KS=0$
Quasi normal if $\Com(A,A^T A)=A(AA^T)-〖(A〗^T A)A=0$
Hyponormal if $\Com(A)=AA^T-A^T A>0$