THE GENERAL COMMON EXACT SOLUTIONS OF COUPLED LINEAR MATRIX AND MATRIX DIFFERENTIAL EQUATIONS

The 2nd International Conference on Research and Education in Mathematics (ICREM 2) – 2005 75

THE GENERAL COMMON EXACT SOLUTIONS OF COUPLED LINEAR MATRIX AND MATRIX DIFFERENTIAL EQUATIONS

Adem Kilicman and Zeyad Abdel Aziz Al Zhour

Department of Mathematics and Institute for Mathematical Research University Putra Malaysia (UPM), Malaysia

43400 , Serdang , Selangor , Darul Ehsan , Malaysia. Email: [email protected] , [email protected]

Abstract: System of matrix and matrix differential equations play an important role in system theory, control theory, stability theory of differential equations communication systems and many other subjects. In this paper, based on the connection between the Kronecker products and Vector- operation, we present the general common exact solutions of coupled linear matrix equations

EYBBXAA =+ 2121 ,

FYDDXCC =+ 2121 and coupled linear matrix differential equations

DtCYBtAXtX )()()(' += ,

BtAYDtCXtY )()()(' += . Several special cases of these systems are also considered; we prove the existence and uniqueness solution of each case, which includes the well-known coupled Sylvester matrix equations, as a special case.

Keywords: Kronecker product (sum), Vector – operator, Matrix equations, Matrix differential equations, Sylvester Equation, Exponential matrix, , Cosh . )(ASinh )(A

1. Introduction There are many books and papers, e.g. [3, 4, 5, 7, 8] devoted to the analysis, existence, uniqueness and representation of the solution and also to the numerical algorithm to solve various types of matrix equations. Most of the existing results, however, are connected with particular systems of such matrix and matrix differential equations. Systems of matrix and matrix differential equations have been widely used in stability theory of differential equations, in the perturbation analysis of linear and non-linear matrix equations and other fields of pure and applied mathematics and also recently in the context of the analysis and numerical simulation of descriptor systems.

In matrix theory, the Kronecker product and vector operator affirming their capability

of solving some matrix equations and matrix differential equations.

In the recent paper, we use the Kronecker products of matrices and vector operator to solve the coupled matrix and matrix differential equations. This method is easy to understand, simple to use and accurate. Furthermore, we discuss several special cases of these systems, and then we prove the existence and uniqueness of the solution of each case. The need to compute thee , cosh( and sinh( are due its appearance in the solutions of coupled matrix and matrix differential equations.

A )A )A

We use the following notations:

• the derivative of matrix function and −)(' tX )(tX −I the identity matrix..

mailto:[email protected]


• the set of all m matrices over −nmM , n× M and when m = n , we write instead of .

nM

nmM ,

• , , Vec , TA 1−A )(A )(Aσ , tr , det( , )(A )A −)(Arank the transpose, inverse of vector operator, eigenvalues, trace, determinant and rank of matrix A , respectively.

2. Some Notations and Preliminary Results

Consider matrices ( ) mij MaA ∈= , ( ) nkl MbB ∈= . Let ( ) mnijij MBaBA ∈=⊗ , BA⊕

be the Kronecker product and Kronecker sum of mnM∈nm BIIA ⊗+⊗= )()( A and B ,

respectively, and VecA the ( )1,1 2,,

mT

mmm Maa ∈L22212 ,,,, maaa L1 ,,m L2111 ,,, aaa= L

vector – operator of . The Kronecker product and vector-operator are related by A

(see, eg., [1,5,7, 8])

Vec (1) VecXABAXB T )()( ⊗=

for any matrices , andnmMA ,∈ qpMB ,∈ pnMX ,∈ .

For any compatibly matrices A , B , and , we shall make frequent use the following properties of the Kronecker product, Kronecker sum and vector – operator that will be useful in our investigation of the solution of matrix and matrix differential equations:

C D

(i) BDACDCBA ⊗=⊗⊗ ))(( (2)

(ii) (3) BABA eee ⊗=⊕ )(

(iii) If { miA i ,...,2,1:)( }== λσ are the eigenvalues of and A { }njB j ,...,2,1:)( == µσ are the eigenvalues of B . Then

(i) { }njmiBA ji ,...,2,1,,...,2,1:)( ===⊗ µλσ . (4)

(ii). { }njmiBA ji ,...,2,1,,...,2,1:)( ==+=⊕ µλσ . (5)

(iv) If is analytic function on the region containing the eigenvalues of f A .Then

and )()( AfIAIf ⊗=⊗ IAfIAf ⊗=⊗ )()( . (6)

Some special cases include (iv):

(i) For , we have AeAf =)(

e and e . (7) IAeIA ⊗=⊗ AeIAI ⊗=⊗

(ii) For , we have )sinh()( AAf =

IAIA ⊗=⊗ )sinh()sinh( and sinh( )sinh() AIAI ⊗=⊗ . (8)

(iii) For , we have )cosh()( AAf =

IAIA ⊗=⊗ )cosh()cosh( and )cosh()cosh( AIAI ⊗=⊗ . (9)

Here, ∑∞

=

=

0 !k

kA

kAe , { }AA eeA −−=

21)sinh( and { }AA eeA −+=

21)cosh( . (10)


For any matrix mMA∈ , the spectral representation of and e assures that: Ae At

e and (11) ∑=

=n

i

iTii

A eyx0

λ ∑=

=n

i

tiTii

At ee yx0

λ

where{ }nλλ ,,1 L are the eigenvalues of A ,{ }nxx ,,1 L be the eigenvectors corresponding to the eigenvalues { }nλλ ,,L1 and { be the eigenvectors of matrix . }nyy ,,1 L

TA

3. Coupled Linear Matrix Differential Equations

In this section we use the Kronecker product and −Vec notation to solve the coupled linear matrix differential equations:

, Y (12) DtCYBtAXtX )()()(' += BtAYDtCXt )()()(' +=

Here, , CAAC = DBBD = , EtX =)( 0 , Y Ft =)( 0 , nd are given constant matrices and ,

EDCBA ,,,, a nMF ∈)(tX nMtY ∈)( are the unknown matrices to be solved.

First, let us introduce the simplest forms of linear matrix differential equations:

(i) , (13) ) CtX =)( 0()(' tAXtX =

Here, A and are given constant matrices. The unique solution of the matrix differential equation defined in (13) is given by

nMC ∈

(14) CtX ttAe )0()( −=

(ii) , BtXtAXtX )()()(' += CtX =)( 0 (15)

Here, , and are given constant matrices. If we use the notation, then we get,

nMA∈−

mMB∈ mnMC ,∈Vec

Vec )( )= . (16) =))(( ' tX () tVecXIBAI nT

m ⊗+⊗ ()( tVecXABT ⊕

Let , , )()( '' tVecXtx = ABH T ⊕= )()( tVecXtx = and VecCc = . Then (16) can be written as follows:

, )(.)(' txHtx = ctx =)( 0 (17)

The unique solution of the matrix differential equation defined in (15) is given by

. (18) )()( 00 ..)( ttBttA eCetX −−=

3.1. Theorem: The general exact solution of coupled linear matrix differential equations defined in (12) is given by

{ }VecFttCDVecEttCDtVecX TTttABTe )])([sinh()])([cosh()( 00))(( 0 −⊗+−⊗= −⊗

{ }VecFttCDVecEttCDtVecY TTttABTe )])([cosh()])([sinh()( 00))(( 0 −⊗+−⊗= −⊗

; (19)

(20)

Proof: If we use the Vec notation, we have the following system: −(.)

. (21)

⊗⊗⊗⊗

=

)()(

)()(

'

'

tVecYtVecX

ABCDCDAB

tVecYtVecX

TT

TT


This system has the following solution (by (14))

(22)

=

−

⊗⊗⊗⊗

VecFVecE

tVecYtVecX tt

ABCDCDAB

TT

TT

e)( 0

)()(

Now we will deal with e .

⊗⊗⊗⊗

ABCDCDAB

TT

TT

Note that

⊗⊗

⊗⊗

00

00

CDCD

ABAB

T

T

T

T

⊗⊗

⊗⊗

=AB

ABCD

CDT

T

T

T

00

00

.

Then

⊗⊗⊗⊗

ABCDCDAB

TT

TT

e

⊗⊗

+

⊗⊗

= 00

00

CDCD

ABAB

T

T

T

T

e

⊗⊗

= ABAB

T

T

e 00

e .

⊗⊗0

0CD

CDT

T

But

=

⊗

⊗

⊗⊗

AB

ABAB

AB

T

TT

T

eee

000

0

and

+−

−+

=⊗−⊗⊗−⊗

⊗−⊗⊗−⊗

⊗⊗

22

22)()(

)()(

00

CDCDCDCD

CDCDCDCD

CDCD

TTTT

TTTT

eeee

eeeeT

T

e .

So

⊗⊗⊗⊗

ABCDCDAB

TT

TT

e

=

⊗

⊗

AB

AB

T

T

ee

00

+−

−+

⊗−⊗⊗−⊗

⊗−⊗⊗−⊗

22

22)()(

)()(

CDCDCDCD

CDCDCDCD

TTTT

TTTT

eeee

eeee

+

−

−

+

⊗−⊗⊗

⊗−⊗⊗

⊗−⊗⊗

⊗−⊗⊗

22

22)()(

)()(

CDCDAB

CDCDAB

CDCDAB

CDCDAB

TTT

TTT

TTT

TTT

eeeee

eeeeee

e=

= .

⊗⊗⊗⊗

⊗⊗

⊗⊗

)cosh()sinh()sinh()cosh(

CDeCDCDCD

TABTAB

TABTAB

TT

TT

eee

Now

=

−

⊗⊗⊗⊗

VecFVecE

tVecYtVecX tt

ABCDCDAB

TT

TT

e)( 0

)()(


−⊗−⊗−⊗−⊗=

−⊗−⊗

−⊗−⊗

))(cosh())(sinh())(sinh())(cosh(

0))((

0))((

0))((

0))((

00

00

ttCDettCDttCDttCD

TttABTttAB

TttABTttAB

TT

TT

eee

VecFVecE

(23)

This gives the following general solution:

{ }VecFttCDVecEttCDtVecX TTttABTe )])([sinh()])([cosh()( 00))(( 0 −⊗+−⊗= −⊗ ,

and

{ }VecFttCDVecEttCDtVecY TTttABTe )])([cosh()])([sinh()( 00))(( 0 −⊗+−⊗= −⊗ .

Now we discuss some special cases of Theorem (3.1), as in the following corollaries:

3.2. Corollary: Consider the coupled matrix differential equations:

DtYBtXtX )()()(' += , Y (24) BtYDtXt )()()(' +=

Here, DBBD = , YEX =)0( , F=)0( a , and ndEDB , nMF ∈ . Then the general exact solution of (24) is given by

{ } BteDtFDtEtX )sinh()cosh()( += , { } BteDtFDtEt )cosh()sinh()( +=Y (25)

Proof: Set and t in (19) and (20), we have nICA == 00 =

{ }VecFtIDVecEtIDtVecX nT

nTtIB n

Te ])[sinh(])[cosh()( )( ⊗+⊗= ⊗

= { }VecFItDVecEItDI nT

nT

ntBTe ][sinh][cosh)( ⊗+⊗⊗

{ }VecEItD nTtBTe ⊗cosh= { }VecFItD n

TtBTe ⊗+ sinh

{ }VecEIDt nTBte ⊗])[cosh(= { }VecFIDt n

TBte ⊗+ ])[sinh( BteDtEtX )cosh()( = BteDtF )sinh(+ { } BteDtFDtE )sinh()cosh( += .

Similarly, Y . { } BteDtFDtEt )cosh()sinh()( +=


)()()(' tCYtAXtX += , Y (26) )()()(' tAYtCXt +=

Here, , YCAAC = EX =)0( , F=)0( a , and ndECA , nMF ∈ . Then the general exact solution of (26) is given by

{ }FCtECttX Ate ).sinh().cosh()( += , { }FCtECtt Ate ).cosh().sinh()( +=Y (27)

Proof: By taking nIBD == and 00 =t in (19) and (20), we have

{ }VecFtCIVecEtCItVecX nntAIne ])[sinh(])[cosh()( )( ⊗+⊗= ⊗

= { }VecFCtIVecECtII nnAt

n e )]sinh([)]cosh([ ⊗+⊗⊗

= +⊗⊗ VecECtII nAt

n e ))cosh()(( VecFCtII nAt

n e ))sinh()(( ⊗⊗

= +⊗ VecECtI Atn e )]cosh([ VecFCtI At

n e )]sinh([ ⊗

= +)).cosh(( ECtVec Ate )).sinh(( FCtVec Ate


= +ECtVec Ate ).cosh(( )).sinh( FCtAte

{ }FCtECttX Ate ).sinh().cosh()( +=

Similarly, Y . { }FCtECtt Ate ).cosh().sinh()( +=


)()()(' tYtXtX += , Y (28) )()()(' tYtXt +=

Here, , Y and EX =)0( F=)0( E and nMF ∈ . Then the general exact solution of (28) is given by

{ })sinh()cosh()( tFtEtX te += , { })sinh()sinh()( tFtEt te +=Y (29)

Proof: Set in (25), we have nIBD ==

{ } tInetFtEtX )sinh()cosh()( += { })sinh()cosh( tFtE += nt Ie

= { })sinh()cosh( tFtEte +

Similarly, { })sinh()sinh()( tFtEt te +=Y .


)()()(' tYBtXtX += , Y (30) BtYtXt )()()(' +=

Here, , Y and andEX =)0( F=)0( EB, nMF ∈ . Then the general exact solution of (30) is given by

{ } BtetFtEtX )sinh()cosh()( += , { } BtetFtEt )cosh()sinh()( +=Y (31)

Proof: Follows immediately by setting nID = in (25).

3.6. Corollary: Consider coupled matrix differential equations:

DtYtXtX )()()(' += , Y (32) )()()(' tYDtXt +=

Here, Y and andEX =)0( , F=)0( ED, nMF ∈ . Then the general exact solution of (32) is given by

{ })sinh()cosh()( DtFDtEtX te += , { })cosh()sinh()( DtFDtEet t +=Y (33)

Proof: Follows immediately by setting nIB = in (25).

4. Coupled Linear Matrix Equations

In this section we use the Kronecker product and −Vec notation to solve and prove the existence and uniqueness solution of the coupled linear matrix equations:

EYBBXAA =+ 2121 , C FYDDXC =+ 2121 (34)

Here,C , C , , , are given constant matrices and

1111 CDD = 2222 CDD = 1111 ,,, DCBA EDCBA ,,,, 2222 nMF ∈X , are the unknown matrices . nMY ∈

Knowledge of the Kronecker product and its applications facilitates our analysis of matrix equations, since the Kronecker product can be used to give a convenient representation for linear matrix equations.


The simplest matrix equation is:

CAXB = (35)

Here, , , C are given constant matrices and is the unknown matrix to be solved . The matrix equation in (35) can be written by using the Vec notation as:

nMA∈ mMB∈ mnM ,∈ mnMX ,∈−(.)

VecCVecXABT =⊗ )( (36)

This system has a unique solution if and only if is invertible if and only if and )( ABT ⊗ AB are invertible.

Now if is not invertible, we consider the rank of the augmented matrix: )( ABT ⊗

]:[ VecCABT ⊗VecXABT =⊗ )(

. If this rank is equal to the rank of the matrix ( , then the system has a solution; otherwise the system has no solution.

)ABT ⊗VecC

The matrix equation in (35) can be generalized as:

∑=

=p

iii CXBA

1 (37)

Here, , ni MA ∈ mi MB ∈ ),...,2,1( pi = and mnMC ,∈ are given constant matrices. Similarly, the general matrix equation in (37) can be written by using the −(.)Vec notation as:

VecCVecXABp

ii

Ti =⊗∑

=1)( (38)

The unique solution is obtained if and only if is invertible, otherwise we

consider a rank argument.

∑=

⊗p

ii

Ti AB

1

)(

An important particular case of (37) is the Sylvester matrix equation:

CXBAX =+ (39)

Here, , and C are given constant matrices, in which the Kronecker sum, frequently turns up. By using the

nMA∈ mMB∈ mnM ,∈−(.)Vec notation in (39), we have

VecCVecXABT =⊕ )( (40)

The system in (40) has a unique solution if and only if is invertible, that is if and only if non of the eigenvalues of ( is zero. So Sylvester matrix equation has a unique solution if and only if and (

)( ABT ⊕)ABT ⊕)BA − have no common eigenvalue if and only if

φσσ =−∩ )()( BA . If on the other hand and A )( B− have an eigenvalue in common, then the existence of the solution depends on the rank of the augmented matrix: [ . If the rank of this matrix is equal to the rank of the matrix , then the solutions exist, otherwise they do not.

]:VecCABT ⊕)( ABT ⊕

4.1. Lemma: Let nMA∈ , and mMB∈ mnMC ,∈ and all eigenvalues of and ( are negatives. Then the unique solution of the matrix equation defined in (39) is

A )B−


∫∞

−=0

... dtCX BtAt ee (41)

As a special case of the matrix equation in (39), set AB −= and , then the following matrix equation:

0=C

XAAX = (42)

has an infinity solutions and the solutions are all matrices which commute with A , for example, are solutions of the matrix equation defined in (37). ,....2,1: == kAX k

Another important particular case of the matrix equation defined in (37) is the following one:

XXAAX α=− (43)

Here, and , which has a non-trivial solution if and only if nMA∈ mMB∈

{ })(:) AA i σλα ∈∈ ( ATσ ⊕− ji λλ −= . Hence, the matrix equation defined in (43) has

a non-trivial solution if and only if ji λλα −= for some ji , .

The previous matrix equations have been considered in several references ( see ,e.g., [5, 8, 10] ).

4.2. Lemma: Let and such thatCBA ,, mMD∈ CAAC = . Then T is invertible if

and only if is invertible.

=

DCBA

CBAD −

Proof: First we prove the lemma for the case when is invertible. It easy to verify that A

− −

n

n

ICAI

1

0

DCBA

=

EBA

0 (44)

Here, is called the Schur complement of in BCADE 1−−= A T . Note that

− −

n

n

ICAI

1

0

EBA

0

is invertible with inverse and T is invertible if and

only if is invertible . Then is invertible if and only if

−

n

n

ICAI

1

0

EB

=

DCBA

A0

E is invertible.

In fact, if E is invertible, then is the inverse of matrix .

−−

−−−

1

111

0 EBEAA

EBA

0

Conversely, suppose is invertible with

as its inverse; then

EBA

0

SRQP

EBA

0

SRQP

=

SRQP

EBA

0

=

n

n

II0

0.

So


++ESER

BSAQBRAP

++

=SERBRAQEPBPA

=

n

n

II0

0.

Since and 0=RA A is invertible, then 0=R , nIPAAP == and , so nISEES == PA =−1 and . We conclude that S=−1E T is invertible if and only if

BCAD 1−− is invertible if and only if is invertible (Since ).

BACAAD 1−− CBADBCAAAD −=−= −1

CAAC =

Finally, the general case (when A is not necessarily invertible) follows by the fact that the set of invertible matrices is dense in (see e.g., [9]). nM

4.3. Lemma: Let and such thatCBA ,, mMD∈ DCCD = . Then T is invertible if

and only if is invertible.

=

DCBA

BCAD −

Proof: When is invertible. It easy to verify that D

DCBA

− −

n

n

ICDI

1

0

=

DBF

0 (45)

Here, is the Schur complement of in CBDAF 1−−= D T .

Proof: Follows by the similar argument of the proof in Lemma (4.2).

4.4. Theorem: The coupled linear matrix equations defined in (34) has a unique solution if and only if is invertible. 11221122 CBCBDADA TTTT ⊗−⊗

Proof: By using the Vec notation, we get −

VecEVecYBBVecXAA TT =⊗+⊗ )()( 1212 (46)

VecFVecYDDVecXCC TT =⊗+⊗ )()( 1212 (47)

This system is equivalent to the following one:

=

⊗⊗⊗⊗

VecFVecE

VecYVecX

DDCCBBAA

TT

TT

1212

1212 (48)

The system defined in (48) has a unique solution if and only if is

invertible if and only if is invertible if and only if is invertible (By Lemma (4.3))

⊗⊗⊗⊗

1212

1212

DDCCBBAA

TT

TT

)1C⊗)(())(( 2121212 CBBDDAA TTTT ⊗−⊗⊗

112 CBT ⊗21122 CBDADA TTT −⊗

Now we discuss some special cases of Theorem (4.4)

4.5. Corollary: Let are given constant matrices such thatECBA ,,, , nMF ∈ BAAB = . Then the coupled linear matrix equations

EBYCAXC =+ , FAYCBXC =+ (49)

has a unique solution if and only if ( is invertible. )() 222 BACT −⊗


Proof: Follows immediately by setting =A =1A 1D , =B =1B and 1C=C 222 BCA == 2D= in Theorem (4.4).

4.6. Corollary: Let ndEDCBA ,,,, a nMF ∈ be given constant matrices such that CD DC= . Then the coupled linear matrix equations

EBYAX =+ , CX FDY =+ (50)

has a unique solution if and only if BCAD − is invertible.

Proof: Set , , C , 1AA = 1BB = 1C= 1DD = and nI 2222 DCBA ==== in Theorem

(4.4).Then the system defined in (50) has a unique solution if and only if BCIADI nn ⊗−⊗ is invertible if and only if )( BCADI n −⊗ is invertible if and only if is invertible. BC−AD

4.7. Corollary: Let ndEDCBA ,,,, a nMF ∈ be given constant matrices such that CD DC= . Then the coupled linear matrix equations

EYBXA =+ , FYDXC =+ (51)

has a unique solution if and only if CBDA − is invertible.

Proof: Set , , C , 2AA = 2BB = 3C= 2DD = and nI 1111 DCBA ==== in Theorem

(4.4). Then the system defined in (51) has a unique solution if and only if is invertible if and only if ( is invertible if and

only if is invertible. n

TTn

TT ICBIDA ⊗−⊗CBDA −

nT ICBDA ⊗− )

4.8. Corollary: Let ndEDCBA ,,,, a nMF ∈ be given constant matrices. Then the coupled Sylvester linear matrix equations

EYBAX =+ , CX FYD =+ (52)

has a unique solution if and only if is invertible if and only if the

matrix is non-singular, in this case, the unique solution is given by

CBAD TT ⊗−⊗

⊗⊗⊗⊗

=IDCIIBAI

H T

T

=

−

VecFVecE

HVecYVecX 1 (53)

Proof: Follows immediately by setting 1AA = , 2BB = , 1CC = , 2DD = and

nI 1212 DCBA ==== in Theorem ( 4.4).

4.9. Corollary: Let ndEDCBA ,,,, a nMF ∈ be given constant matrices. Then the coupled linear matrix equations

EBYXA =+ , FDYXC =+ (54)

has a unique solution if and only if is invertible . BCDA TT ⊗−⊗

Proof: Follows immediately by setting 2AA = , 1BB = , 2CC = , 1DD = and

nI 2121 DCBA ==== in Theorem ( 4.4).


4.10. Theorem: Let and ECBA ,,, nMF ∈ be given constant matrices such that BAAB = . Then the coupled linear matrix equations

EBYCAXC =+ , FAYCBXC =+ (55)

has a unique solution if and only if C , BA + and BA − are invertible matrices, in this case, the unique solution is given by

{ 111 )()()()(21 −−− −−+++= CFEBAFEBAX } (56)

{ 111 )()()()(21 −−− −−−++= CFEBAFEBAY } (57)

Proof: By using the Vec notation, the system becomes −

VecEVecYBCVecXAC TT =⊗+⊗ )()(

VecFVecYACVecXBC TT =⊗+⊗ )()(

This system is equivalent to the following one:

=

⊗⊗⊗⊗

VecFVecE

VecYVecX

ACBCBCAC

TT

TT

(58)

The system defined in (58) has a unique solution if and only if is

invertible. One can easily show that:

⊗⊗⊗⊗

ACBCBCAC

TT

TT

⊗⊗⊗⊗

ACBCBCAC

TT

TTT

T

T

UBAC

BACU

−⊗+⊗

=)(0

0)( (59)

Here, U =

−

nn

nn

IIII

21 is a unitary matrix. So

⊗⊗⊗⊗

ACBCBCAC

TT

TT

)( BACT +⊗

is invertible if and only if if and only if

and C are invertible matrices if and only if ,

−⊗+⊗

)(00)(

BACBAC

T

T

C)( BAT −⊗ BA + and BA − are invertible matrices. Note that

1−

⊗⊗⊗⊗

ACBCBCAC

TT

TT1

11

)(00)(

)( −

−

−

−⊗+⊗

= UBAC

BACU

T

TT

TT

T

UBAC

BACU

−⊗+⊗

=−−

−−

11

11

)()(00)()(

− nn

nn

IIII

21

=

−⊗+⊗

−−

−−

11

11

)()(00)()(

BACBAC

T

T

−

nn

nn

IIII

{ } { }{ } {

−++⊗−−+⊗−−+⊗−++⊗

=−−−−−−

−−−−−−

111111

111111

)()()()()()()()()()()()(

21

BABACBABACBABACBABAC

TT

TT

} (60)


Then

⊗⊗⊗⊗

=

−

VecFVecE

ACBCBCAC

VecYVecX

TT

TT 1

(61)

This gives:

{ } { }( )VecFBABACVecEBABACVecX TT 111111 )()()()()()(21 −−−−−− −−+⊗+−++⊗=

{ } { }( )111111 )()()()(21 −−−−−− −−++−++ FCBABAECBABAVec=

{ }1111 )()()()(21 −−−− −−+++ CFEBACFEBAVec=

Hence, { } 111 )()()()(21 −−− −−+++= CFEBAFEBAX .

Similarly, { } 111 )()()()(21 −−− −−−++= CFEBAFEBAY .

4.11. Theorem: Let and ECBA ,,, nMF ∈ be given constant matrices such that BAAB = . Then the coupled linear matrix equations

ECYBCXA =+ , CXB FCYA =+ (62)

has a unique solution if and only if C , BA + and BA − are invertible matrices, in this cas, the unique solution is given by

{ })))(())((21 111 −−− −−+++= BAFEBAFECX (63)

{ })))(())((21 111 −−− −−−++= BAFEBAFECY (64)

Proof: Follows by the similar argument of the proof in Theorem (4.10).

4.12. Corollary: Let and EBA ,, nMF ∈ be given constant matrices such that BAAB = . Then the coupled linear matrix equations

EBYAX =+ , FAYBX =+ (65)

has a unique solution if and only if BA + and BA − are invertible matrices, in this case, the unique solution is given by

{ )()()()(21 11 FEBAFEBAX −−+++= −− } (66)

{ )()()()(21 11 FEBAFEBAY −−−++= −− } (67)

Proof: Follows immediately by setting C nI= in (56) and (57).

4.13. Corollary: Let and EBA ,, nMF ∈ be given constant matrices such that BAAB = . Then the coupled linear matrix equations

EYBXA =+ , FYAXB =+ (68)


has a unique solution If and only if BA + and BA − are invertible matrices, in this case, the unique solution is given by

{ )))(())((21 11 −− −−+++= BAFEBAFEX } (69)

{ )))(())((21 11 −− −−−++= BAFEBAFEY } (70)

Proof: Follows immediately by setting nIC = in (63) and (64).

Acknowledgment This research is supported by University Putra Malaysia (UPM) under the Grant IRPA 09-02-04-0259-EA001.

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THE GENERAL COMMON EXACT SOLUTIONS OF COUPLED LINEAR MATRIX AND MATRIX DIFFERENTIAL EQUATIONS

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