In the previous two sections, two types of homogeneous complex conjugate matrix equations are investigated based on unilateral linear matrix equations. In this section, the unilateral-equation-based approach is used to solve the following nonhomoge- neous con-Sylvester matrix equation
AX+BY =XF+R, (8.32)
where A∈Cn×n,B∈Cn×r,F∈Cp×p, andR∈Cn×p are given matrices, andX∈ Cn×pandY ∈Cr×pare unknown matrices to be determined. WhenR=0, this equa- tion degenerates into the following form
AX+BY =XF. (8.33)
This is the homogeneous con-Sylvester matrix equation studied in Sects.6.3,7.1, and8.1. Before proceeding, the following proposition is needed. The result of this proposition can be easily derived, and has been given in Sect.7.1.
Proposition 8.1 If (Xs,Ys) is a special solution to the nonhomogeneous con- Sylvester matrix equation (8.32) and(X∗,Y∗)is the general solution to the con- Sylvester matrix equation (8.33), then the general solution to the nonhomogeneous con-Sylvester matrix equation (8.32) is given by
X=Xs+X∗,Y =Ys+Y∗.
The above result can be viewed as superposition principle for con-Sylvester matrix equation. According to this principle, in order to solve the nonhomogeneous con- Sylvester equation (8.32), one needs to solve the corresponding homogeneous equa- tion (8.33) and obtain a special solution of the equation itself. In Sect.8.1, the general solution to the equation (8.33) has been provided. Thus, in the sequel the emphasis is placed on obtaining a special solution to the nonhomogeneous con-Sylvester matrix equation (8.32). It should be pointed that the same symbols as in Sect.8.1are used.
Theorem 8.5 Given matrices A∈Cn×n, B∈Cn×r, F∈Cp×p, and R∈Cn×p, if there exist two integers τ and q, a matrix setDs=
Dsi ∈Cr×q, i∈I[1, τ] and two matrices V ∈Cn×qand W ∈Cq×psuch that
BDs0+ABD1s+AABDs2+ ã ã ã +A−→τ BDsτ∗τ
=V (8.34)
and
V W =R, (8.35)
then a special solution to the nonhomogeneous con-Sylvester matrix equation (8.32) can be given by
Xs=−→
Ctrτ(A,B)Sτ(Ds)←−
Obsτ(F,W) Ys=
Ds0Ds1ã ã ãDsτ←−
Obsτ+1(F,W) . (8.36)
Proof By virtue of (8.34), a direct computation gives AXs
=A−→
Ctrτ(A,B)Sτ(Ds)←−
Obsτ(F,W)
=
AB A−→2Bã ã ãA−−→τ−1B∗(τ−1)A−→τB∗τ
Sτ(Ds)←−
Obsτ(F,W)
= τ
i=1
A−→i(BDsi)∗i τ−1
i=1
A−→i(BDsi+1)∗i ã ã ãABDsτ ←−
Obsτ(F,W)
=
−BDs0 τ−1
i=1
A−→i(BDsi+1)∗i ã ã ãABDsτ ←−
Obsτ(F,W)+V W
=
−BDs0 τ−1
i=1
A−→i(BDsi+1)∗i ã ã ãABDsτ 0 ←−
Obsτ+1(F,W)+V W. Similarly, it can be derived that
XsF=−→
Ctrτ(A,B)Sτ(Ds)←−
Obsτ(F,W)F
=−→
Ctrτ(A,B)
0Sτ(Ds)←−
Obsτ+1(F,W)
=
0τ−1
i=0
A−→i(BDsi+1)∗iã ã ãBDsτ ←−
Obsτ+1(F,W). It follows from the previous two relations and (8.35) that
AXs−XsF=
−BDs0 −BDs1 ã ã ã −BDsτ←−
Obsτ+1(F,W)+V W
= −B
Ds0Ds1ã ã ãDsτ←−
Obsτ+1(F,W)+R
= −BYs+R.
This shows that the matrix pair(Xs,Ys)given in (8.36) satisfies the nonhomogeneous
con-Sylvester matrix equation (8.32).
With Proposition8.1, one can obtain the general solution to the nonhomogeneous con-Sylvester matrix equation (8.32) according to the results of Lemma 8.2and Theorem8.5.
Theorem 8.6 Given matrices A∈Cn×n, B∈Cn×r,F∈Cp×p, and R∈Cn×p, sup- pose that λ(AA)∩λ(FF)=∅. If there exist integers t, τ and q, a matrix set Ds=
Dsi ∈Cr×q, i∈I[1, τ]
and two matrices V ∈Cn×q and W ∈Cq×p satis- fying (8.34) and (8.35), and a matrix setD=
Di∈Cr×r, i∈I[1,t]
satisfying (8.11), then the matrices X and Y given by
X=−→
Ctrt(A,B)St(D)←−
Obst(F,Z)+−→
Ctrτ(A,B)Sτ(Ds)←−
Obsτ(F,W) Y =
D0D1 ã ã ãDt
←−
Obst+1(F,Z)+
Ds0Ds1ã ã ãDsτ←−
Obsτ+1(F,W) (8.37) with Z∈Cr×pbeing an arbitrarily chosen free parameter matrix, satisfy the matrix equation (8.32). Further, defineD(s)as in (8.7). Then all the solutions to the homo- geneous con-Sylvester matrix equation (8.32) can be parameterized by (8.37) if
detD(s) =0, for any s∈λ (Fσ).
In the above theorem, the solution to the nonhomogeneous con-Sylvester matrix equation (8.32) is expressed in terms of the con-controllability matrix associated with (A,B), and the con-observability matrices. This property may bring convenience and advantages to the further analysis of the matrix equation (8.32).
When the approach in Theorem8.6is adopted to solve the con-Sylvester matrix equation (8.32), two groups of matricesDi∈Cr×r,i∈I[1,t], andDsi ∈Cr×q,i∈ I[1, τ], are required. They need to satisfy (8.11) and (8.34), respectively. In Sect.8.1, it has been illustrated that the matrix group
Di∈Cr×r,i∈I[1,t]
can be obtained by the standard unilateral matrix equation (8.18). Such an approach can also be applied to derive the matrix group
Dsi ∈Cr×q,i∈I[1, τ]
satisfying (8.34). In fact, by letting
Ls=
⎡
⎢⎢
⎣ Ds0 Ds1
ã ã ã Dst∗τ
⎤
⎥⎥
⎦,
the expression (8.34) can be equivalently written as
−→Ctrτ+1(A,B)Ls=V. (8.38)
This is a standard unilateral matrix equation, and can be solved easily. One possible method to solve it is to use numerically reliable singular value decompositions.
Example 8.3 Consider a nonhomogeneous con-Sylvester matrix equation in the form of (8.32) with the following parameters
A=
⎡
⎣1−2−i−1+i
0 i 0
0 −1 1−i
⎤
⎦, F =
2i i 1 −1+i
,
B=
⎡
⎣−1+i 1
0 i
−i 1−2i
⎤
⎦, R=
⎡
⎣1 i i 1 0 1−i
⎤
⎦.
The corresponding homogeneous equation has been considered in Sect.8.1. For this matrix equation, n=3. The integerτ is chosen to beτ =2. A direct calculation gives
B AB AAB
=
⎡
⎣−1+i 1 −2−2i−3+i−2+4i−6+3i
0 i 0 1 0 i
−i 1−2i 1+i 3+2i −2i −5i
⎤
⎦.
In addition, the matricesW andV are respectively chosen asW =I2 andV =R.
With this, by solving the matrix equation (8.38) one can obtain a group of matrices
satisfying (8.34) as follows:
Ds0=
−2−2i 1+6i
1 0
,Ds1=
3
2 −12i−2+2i
0 0
,D2s =
1−2 0 −i
. Furthermore, it can be derived that
−−→Ctr2(A,B)= B AB
,S2(Ds)=
Ds1Ds2 Ds2 0
,
←−−Obs2(F,W)= W
W F
,←−−
Obs3(F,W)=
⎡
⎣ W W F W FF
⎤
⎦.
According to (8.36), a special solution to the nonhomogeneous con-Sylvester matrix equation in this example can be given by
Xs =−−→
Ctr2(A,B)S2(Ds)←−−
Obs2(F,W)
=
⎡
⎣ 1+i −1+i
1 −1
−3.5−0.5i −4i
⎤
⎦, (8.39)
Ys=
Ds0Ds1 D2s←−−
Obs3(F,W)
=
3 0.5+2.5i 2+i 1−2i
. (8.40)
By using Theorem8.6and the result in Sect.8.1, the general solution to the homo- geneous con-Sylvester matrix equation can be expressed as
X =N0Z+N1ZF+Xs
Y =D0Z+D1ZF+D2ZFF+Ys ,
whereXsandYsare respectively given in (8.39) and (8.40), and the matricesN0,N1, D0,D1, andD2are given as follows
D0=
−1+i−12+6i
0 −2
,D1=
−i−5+i
0 0
,D2=
1 0 0 2
,
N0=
⎡
⎣−1+i−2+4i
0 2
−i 7−9i
⎤
⎦,N1=
⎡
⎣−1−i 2
0 −2i
i 2+4i
⎤
⎦.
At the end of this section, two remarks are provided for the proposed approaches.
Remark 8.9 When Theorem8.5is utilized to obtain a special solution for the non- homogeneous con-Sylvester matrix equation (8.32), a group of matricesDsi ∈Cr×q, i∈I[1, τ], need to be chosen such that the condition (8.34) holds. As mentioned pre- viously, these matrices can be obtained from the unilateral matrix equation (8.38).
A necessary condition for the solvability of (8.38) is rank
−→
Ctrτ+1(A,B)
=rank −→
Ctrτ+1(A,B)V
.
Therefore, it is important to determine an integerτ such that this condition holds.
However, it is a pity that no method is provided in this book to determine such aτ. Interested readers are encouraged to investigate this problem.
Remark 8.10 In this section, the unilateral-equation-based approach is proposed to solve the nonhomogeneous con-Sylvester matrix equation. In fact, such a method can also be generalized to solve the following nonhomogeneous con-Yakubovich matrix equation
X−AXF=BY +R,
where A∈Cn×n,B∈Cn×r,F∈Cp×p, andR∈Cn×p are given matrices, andX∈ Cn×pandY ∈Cr×p are unknown matrices to be determined. With the idea in this section, the special and general solutions of this equation can be easily derived.