More on the conjecture by Šiljak

Let
\tilde{a}
be a special class of matrices with complex elements. This correspondence considers the properties of any
A \in \tilde{a}
which will guarantee that if
G = - (A*H + HA)
, then for any given Hermitian positive definite matrix
H
, there exists a unique nonsingular Hermitian matrix
G
. Properties of the eigenvalues of
A
and
G
are established.