Matrices satisfying Siljak's conjecture

Siljak's conjecture on the existence of a symmetric positive definite matrix
H
having a specified structure and satisfying matrix equation
HA + A^{T}H = - W
, is shown to be true in cases when
A
is a companion matrix of a polynomial,
A
is a Schwarz matrix and
A
is a nonderogatory matrix in its Routh-canonical form, by actually constructing the matrix
H
in each case.