A=((A+A^T))/2+((A-A^T))/2=S+K,with S=((A+A^T))/2 and K=((A-A^T))/2

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 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)

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