Abstract
This paper considers a new method to solve the first-order and second-order nonhomogeneous generalized Sylvester matrix equations
A
V
+
B
W
=
E
V
F
+
R
and
M
V
F
2
+
D
V
F
+
K
V
=
B
W
+
R
, respectively, where
A
,
E
,
M
,
D
,
K
,
B
, and
F
are the arbitrary real known matrices and
V
and
W
are the matrices to be determined. An explicit solution for these equations is proposed, based on the orthogonal reduction of the matrix
F
to an upper Hessenberg form
H
. The technique is very simple and does not require the eigenvalues of matrix
F
to be known. The proposed method is illustrated by numerical examples.