This section is devoted to investigate the following homogeneous con-Sylvester matrix equation
AX+BY =XF, (7.2)
where A∈Cn×n, B∈Cn×r, and F∈Cp×p are known matrices, andX andY are the matrices to be determined. Before proceeding, the following result on normal con-Sylvester matrix equations is given.
Lemma 7.2 Given matrices A∈Cn×n and F∈Cp×p, the normal con-Sylvester matrix equation
XF−AX=C (7.3)
has a unique solution for any C ∈Cn×pif and only ifλ(AA)∩λ(FF)=∅.
The result of Lemma7.2is in fact the Theorem6.3in the previous section.
Based on this lemma, it is known that the con-Sylvester matrix equation (7.2) has a unique solutionX with respect to an arbitrary fixed matrixY if the matrices AA andFF have no common eigenvalues. Therefore, when λ(AA)∩λ(FF)=∅, the number of degrees of freedom in the solution(X,Y)to the matrix equation (7.2) is equal to the number of elements in the matrixY, that is,rp. This fact is expressed in the following lemma.
Lemma 7.3 Given matrices A∈Cn×n, B∈Cn×r, and F∈Cp×p, if λ(AA)∩λ (FF)=∅, then the number of degrees of freedom existing in the solution(X,Y) to the homogeneous con-Sylvester matrix equation (7.2) is rp.
With this lemma, a basic solution can be obtained for the homogeneous con- Sylvester matrix equation (7.2).
Theorem 7.1 Given matrices A∈Cn×n, B∈Cn×r, and F∈Cp×p, suppose that λ(AA)∩λ(FF)=∅. Let
fAA(s)=det(sI−AA)=
n−1
i=0
αisi, αn=1,
adj(sI−AA)=
n−1
i=0
Risi.
Then, all the solutions to the homogeneous con-Sylvester matrix equation (7.2) can be characterized by
⎧⎪
⎨
⎪⎩ X =
n−1
i=0
RiBZF(FF)i+A
n−1
i=0
RiBZ(FF)i Y =ZfAA(FF)
, (7.4)
where Z ∈Cr×pis an arbitrarily chosen free parameter matrix.
Proof It is first shown that the matricesX andY given in (7.4) are solutions to the con-Sylvester matrix equation (7.2). A direct calculation gives
AX−XF
=A
n−1
i=0
RiBZF(FF)i+AA
n−1
i=0
RiBZ(FF)i
−
n−1
i=0
RiBZF(FF)iF−A
n−1
i=0
RiBZ(FF)iF (7.5)
=AA
n−1
i=0
RiBZ(FF)i−
n−1
i=0
RiBZ(FF)i+1
=AAR0BZ+
n−1
i=1
(AARi−Ri−1)BZ(FF)i−Rn−1BZ(FF)n.
By the following polynomial matrix relation
(sI−AA)adj(sI−AA)=Idet(sI−AA), it is easily derived that
⎧⎨
⎩
−AAR0=α0I,
−AARi+Ri−1=αiI, i∈I[1,n−1], Rn−1=αnI.
(7.6)
By applying these relations to (7.5) one has AX−XF
= −α0BZ−
n−1
i=1
αiBZ(FF)i−αnBZ(FF)n (7.7)
= −BZ n
i=0
αi(FF)i
= −BZfAA(FF).
Therefore, the matrices X andY given in (7.4) satisfy the con-Sylvester matrix equation (7.2).
Secondly, let us show the completeness of solution (7.4). Sinceλ(AA)∩λ(FF)=
∅andλ(FF)=λ(FF), the matrixfAA(FF)is nonsingular, and hence the mapping Z →Y given in (7.4) is injective. This implies that the mappingZ →(X,Y)given in (7.4) is also injective. So all therpelements inZhave contribution to the solution (7.4). In addition, it follows from Lemma7.3that there arerpdegrees of freedom in
the solution of the con-Sylvester matrix equation (7.2). These two facts imply that
the solution given in (7.4) is complete.
The above theorem provides a very neat closed-form solution to the homogeneous con-Sylvester matrix equation (7.2). In order to obtain this solution, one needs the coefficientsαi,i∈I[1,n], of the polynomialfAA(s)and the coefficient matricesRi, i∈I[0,n−1], of adj(sI−AA). They can be obtained by the following Leverrier algorithm in Sect.2.2for the matrixAA:
Rn−i=Rn−i+1AA+αn−i+1I, Rn=0
αn−i= −tr(Rn−iiAA), αn=1 ,i∈I[1,n]. (7.8) In fact, during the proof of Theorem 7.1the relation (7.6) is the first expression of the Leverrier algorithm (7.8). The iterative expression in (7.8) can be explicitly expanded as
⎧⎪
⎪⎪
⎪⎪
⎨
⎪⎪
⎪⎪
⎪⎩
R0 =α1In+α2AA+ ã ã ã +αn−1 AAn−2
+ AAn−1
R1=α2In+α3AA+ ã ã ã + AAn−2
ã ã ã
Rn−2=αn−1In+AA Rn−1=In
. (7.9)
From these relations it is easily known that R0B R1Bã ã ãRn−1B
=Ctr(AA,B)Sn(fAA(s)Ir). Thus one has
n−1
i=0
RiBZF(FF)i
=
R0B R1Bã ã ãRn−1B
Obsn(FF,ZF)
=Ctr(AA,B)Sn(fAA(s)Ir)Obsn(FF,ZF). Similarly, it can be obtained that
n−1
i=0
RiBZ(FF)i
=Ctr(AA,B)Sn(fAA(s)Ir)Obsn(FF,Z).
With the above discussion one can obtain an equivalent form of the solution provided in Theorem7.1.
Theorem 7.2 Given matrices A∈Cn×n, B∈Cn×r, and F∈Cp×p, suppose that λ(AA)∩λ(FF)=∅. Then all the solutions to the con-Sylvester matrix equation (7.2) can be characterized by
⎧⎪
⎨
⎪⎩
X =Ctr(AA,B)Sn(fAA(s)Ir)Obsn(FF,ZF) +ACtr(AA,B)Sn(fAA(s)Ir)Obsn(FF,Z) Y =ZfAA(FF)
, (7.10)
with Z being an arbitrarily chosen parameter matrix.
If Theorem7.2is applied to obtain the general solution of the con-Sylvester matrix equation (7.2), one needs only the coefficients αi,i∈I[0,n], of the characteristic polynomial ofAA. These coefficients can be obtained by some proper numerically reliable algorithms (e.g. [189]) apart from the Leverrier algorithm (7.8). In the above theorem, the solution of the homogeneous con-Sylvester matrix equation (7.2) is expressed in terms of the controllability matrix associated with the matricesAAand B, and the observability matrix associated with the matrixFand the free parameter matrixZ. This property may bring convenience and advantages to the further analysis of the con-Sylvester matrix equation (7.2).
By applying the results in Theorems7.1and7.2, explicit solutions for the normal con-Sylvester matrix equation (7.3) can be obtained. In addition, it is easily found that the matrix equation (7.2) is reduced to the normal con-Sylvester matrix equation (7.3) ifB=CandY =I. With such an observation, the solutionX of the normal con-Sylvester matrix equation (7.3) can be obtained from the first expression in (7.4) or (7.10) by lettingB=Cand setting the free parameterZto be
Z=
fAA(FF)−1 . This is the following corollary.
Corollary 7.1 Given matrices A∈Cn×n, B∈Cn×r, and F∈Cp×p, suppose that λ(AA)∩λ(FF)=∅. Let
fAA(s)=det(sI−AA)= n
i=0
αisi, αn=1,
adj(sI−AA)=
n−1
i=0
Risi.
Then the unique solution of the normal con-Sylvester matrix equation (7.3) is given by
X=
n−1
i=0
RiC
fAA(FF)−1
F(FF)i+A
n−1
i=0
RiC
fAA(FF)−1
(FF)i, (7.11)
or equivalently
X =Ctr(AA,C)Sn(fAA(s)Ir)Obsn(FF,
fAA(FF)−1
F) (7.12)
+A
Ctr(AA,C)Sn(fAA(s)Ir)Obsn(FF,
fAA(FF)−1
)
. It is easily checked that
F(FF)ifAA(FF)=fAA(FF)F(FF)i, which, under the conditionλ(AA)∩λ(FF)=∅, is equivalent to
fAA(FF)−1
F(FF)i=F(FF)i
fAA(FF)−1
. (7.13)
From Lemma2.19, it is known thatfAA(s)∈R[s]. Combining this fact with (7.9), gives
ARi=RiA. (7.14)
In addition, due tofAA(s)∈R[s]it is easily derived that fAA(FF)−1
=
fAA(FF)−1
=
fAA(FF)−1
. With this relation, it can be obtained that
fAA(FF)−1
(FF)i=(FF)i
fAA(FF)−1
. (7.15)
By combining the three relations (7.13)–(7.15) it is known that the expression (7.11) for the unique solution of the matrix equation (7.3) can be equivalently rewritten as
X= n−1
i=0
RiCF(FF)i+
n−1
i=0
RiAC(FF)i
fAA(FF)−1 .
This is exactly the result given in Theorem6.7. Different from the approach in the current section, in Chap.6this solution was derived by combining a solution of a normal Sylvester matrix equation and some properties of the real representation of a complex matrix.
At the end of this section, the solution of the following Sylvester matrix equation is discussed
AX+BY =XF, (7.16)
whereA∈Cn×n,B∈Cn×r, andF∈Cp×pare known matrices, by applying the pro- posed results on the con-Sylvester matrix equation. If the known matricesA,B, and F, and the unknown matricesX andY are all real, then the matrix equation (7.2) becomes the matrix equation (7.16) withA∈Rn×n,B∈Rn×r, andF∈Rp×pbeing known matrices, andX ∈Rn×pandY ∈Rr×pbeing the unknown matrices. By apply- ing Theorem7.1, under the condition ofλ(A)∩λ(F)=∅all the real solutions of the Sylvester matrix equation (7.16) can be parameterized as
⎧⎪
⎨
⎪⎩ X=
n−1
i=0
RiBZF2i+1+A
n−1
i=0
RiBZF2i Y =ZfA2(F2)
, (7.17)
where n−1
i=0 Risi=adj(sI−A2),
andZ ∈Rr×pis an arbitrarily chosen parameter matrix representing the degrees of freedom in the solution. In view that the set of complex numbers with addition and multiplication is a field, the result in (7.17) holds for the Sylveter matrix equation (7.16) with complex known and unknown matrices. This can be summarized as the following corollary.
Corollary 7.2 Given matrices A∈Cn×n, B∈Cn×r, and F∈Cp×p, suppose that λ(A)∩λ(F)=∅. Let
adj(sI−A2)=
n−1
i=0
Risi. (7.18)
Then all the solutions(X, Y)to the Sylvester matrix equation (7.16) can be parame-
terized as ⎧
⎪⎨
⎪⎩ X=
n−1
i=0
(RiBZF+ARiBZ)F2i Y =ZfA2(F2)
,
or equivalently ⎧
⎨
⎩
X =Ctr(A2,B)Sn(fA2(s)Ir)Obsn(F2,ZF) +ACtr(A2,B)Sn(fA2(s)Ir)Obsn(F2,Z)
Y =ZfA2(F2) ,
where Z ∈Cr×pis an arbitrarily chosen free parameter matrix.
Similarly to the idea of Corollary 7.1, from Corollary7.2one can also obtain explicit expressions of the unique solution of the normal Sylvester matrix equation
XF−AX=C (7.19)
with A∈Cn×n,C∈Cn×p, and F∈Cp×p being known matrices. In addition, the solution of (7.19) can also be obtained by specializing the solution to the equation (7.3). With these ideas, the following corollary is given.
Corollary 7.3 Given matrices A∈Cn×n, C∈Cn×p, and F∈Cp×p, suppose that λ(A)∩λ(F)=∅, and (7.18) holds. Then the unique solution of the normal Sylvester matrix equation (7.19) can be expressed by one of the following
X =
n−1
i=0
RiC
fA2(F2)−1
F+ARiC
fA2(F2)−1 F2i,
X = n−1
i=0
Ri(CF+AC)F2i
fA2(F2)−1 ,
X = n−1
i=0
(RiCF+ARiC)F2i
fA2(F2)−1 , X =Ctr(A2,C)Sn(fA2(s)Ir)Obsn(F2,
fA2(F2)−1
F) +ACtr(A2,C)Sn(fA2(s)Ir)Obsn(F2,
fA2(F2)−1 ), X =
Ctr(A2,C)Sn(fA2(s)Ir)Obsn(F2,F)+
ACtr(A2,C)Sn(fA2(s)Ir)Obsn(F2,Ip) fA2(F2)−1
, X=
Ctr(A2,C)Sn(fA2(s)Ir)Obsn(F2,F)+
Ctr(A2,A)Sn(fA2(s)Ir)Obsn(F2,C) fA2(F2)−1
.
In [300], for the Sylvester matrix equation (7.16) the following two explicit expres- sions of the general solution were given under the conditionλ(A)∩λ(F)=∅
⎧⎪
⎨
⎪⎩ X =
n−1
i=0
R˜iBZF Y =ZfA(F)
,
and
X=Ctr(A,B)Sn(fA(s)Ir)Obsn(F,Z)
Y =ZfA(F) ,
where n−1
i=0 R˜isi=adj(sI−A), and Z ∈Cr×p is an arbitrarily chosen parameter matrix representing the degrees of freedom in the solution. In addition, in [258] the following two expressions for the unique solution to the normal Sylvester matrix equation (7.19) withλ(A)∩λ(F)=∅were also provided
X= n−1
i=0
R˜iCF
fA(F)−1
(7.20) and
X=Ctr(A,C)Sn(fA(s)Ir)Obsn(F,I)
fA(F)−1
, where
n−1
i=0
R˜isi=adj(sI−A).
Clearly, the expressions proposed in Corollaries7.2and7.3are more complicated than those in [258, 300], respectively. Moreover, the solutions in Corollaries7.2and 7.3are dependent on the information of matrixA2instead ofA. Nevertheless, the approach in the current section may provide a new idea to investigate the matrix equations (7.2) and (7.19). It is desired that the proposed results in this section can provide some directions to further exploit some properties of these matrix equa- tions. In addition, it is very interesting to show the equivalence of the expressions in Corollary7.3and that in (7.20) by a direct computation.