Miskolc Mathematical Notes
Vol. 17 (2016), No. 1, pp. 635–645
HU e-ISSN 1787-2413
DOI: 10.18514/MMN.2016.1435
ON THE MATRIX NEARNESS PROBLEM FOR
(SKEW–)SYMMETRIC MATRICES ASSOCIATED WITH THE
MATRIX EQUATIONS .A1 XB1 ; : : : ; Ak XBk / D .C1 ; : : : ; Ck /
S. S.İMS.EK, M. SARDUVAN, AND H. ÖZDEMİR
Received 19 November, 2014
Abstract. Suppose that the matrix equations system .A1 XB1 ; : : : ; Ak XBk / D .C1 ; : : : ; Ck / with
unknown matrix X is given, where Ai , Bi , and Ci , i D 1; 2; : : : ; k; are known matrices of suitable
sizes. The matrix nearness problem is considered over the general and least squares solutions of
this system. The explicit forms of the best approximate solutions of the problems over the sets of
symmetric and skew–symmetric matrices are established as well. Moreover, a comparative table
depending on some numerical examples in the literature is given.
2010 Mathematics Subject Classification: 15A06; 15A09; 15A24; 65F35
Keywords: best approximate solution, Frobenius norm, matrix equations, Moore–Penrose generalized inverse, least squares solutions
1. I NTRODUCTION AND N OTATIONS
Let Rm;n , Rn , RnS , and RnSS be the sets of m n real matrices, n n real
matrices, n n real symmetric matrices, and n n real skew–symmetric matrices,
respectively. The symbols AT and A will denote the transpose and the Moore–
Penrose generalized inverse of a matrix A 2 Rm;n , respectively. Further, vec ./
T
will stand for the vec operator, i.e., vec .A/ D a1T ; a2T ; : : : ; anT
for the matrix
A D .a1 ; a2 ; : : : ; an / 2 Rm;n , ai 2 Rm;1 , i D 1; 2; : : : ; n; and A ˝ B will stand for
the Kronecker product of matrices A 2 Rm;n and B 2 Rp;r (see [1]).
Moreover, let
Rm1 ;n1 Rmk ;nk D fŒA1 ; : : : ; Ak j Ai 2 Rmi ;ni ; i D 1; 2; : : : ; kg:
It is easy to see that Rm1 ;n1 Rmk ;nk is a linear space over the real number
field. We define the inner product for all ŒA1 ; A2 ; : : : ; Ak and ŒB1 ; B2 ; : : : ; Bk 2
Rm1 ;n1 Rm2 ;n2 Rmk ;nk in the linear space as follows:
hŒA1 ; A2 ; : : : ; Ak ; ŒB1 ; B2 ; : : : ; Bk i D t r.B1T A1 / C C t r.BkT Ak /;
This work was supported by Research Fund of the Sakarya University. Project Number: 2014-0200-001.
c 2016 Miskolc University Press
636
S. S.İMS.EK, M. SARDUVAN, AND H. ÖZDEMİR
then Rm1 ;n1 Rm2 ;n2 Rmk ;nk is a Hilbert inner space. Furthermore, let kkH
denotes the norm that is derived by inner product, i.e.,
kŒA1 ; A2 ; : : : ; Ak kH D hŒA1 ; A2 ; : : : ; Ak ; ŒA1 ; A2 ; : : : ; Ak i1=2
D Œt r.AT1 A1 / C t r.AT2 A2 / C C t r.ATk Ak /1=2
1=2
D kA1 k2 C : : : C kAk k2
;
where kk denotes the Frobenius norm (see, for example, [11]).
The well–known linear matrix equation AXB D C , where A, B, C are known
matrices of suitable sizes and X is the matrix of unknowns, were studied in the case
of special solution structures, e.g. symmetric, triangular, Hermitian, nonnegative
definite, reflexive, diagonal etc. using matrix decomposition such as the singular
value decomposition (SVD), the generalized SVD (GSVD), the quotient SVD, and
the canonical correlation decomposition (CCD) in [5, 6, 12, 15–17, 23, 36].
Now, first, consider the following two problems.
Problem 1. For given matrices A 2 Rm;n , B 2 Rn;r , and C 2 Rm;r , find XO 2 ˝
such that
AXO B C D min kAXB C k ;
X 2˝
where ˝ is anyone of the sets of special matrices such as symmetric, skew–symmetric,
Hermitian, reflexive etc.
Problem 2. Let SE1 be the solution set of Problem 1. For a given matrix X0 2 Rn ,
find XO 2 SE1 such that
XO
X0 D min kX
X2SE1
X0 k :
Problem 2 which is very important in applied sciences is known as the matrix
nearness problem in the literature and it has been extensively studied in recent years.
For instance, Peng et al. [28] and Huang et al. [14] presented matrix iteration methods for finding the symmetric and skew–symmetric solutions of Problem 2, respectively. Peng et al. [27] gave the necessary and sufficient conditions for solvability of
Problem 2 over reflexive and anti–reflexive matrices. Moreover, they obtained the
explicit expression of the optimal approximation solution of Problem 2 when X is a
reflexive and an anti–reflexive matrix. In these literatures, the linear matrix equation
AXB D C is consistent. But, it is rarely possible to satisfy the consistency condition
of the linear matrix equation AXB D C , since the matrices A, B, and C occurring
in practice are usually obtained from an experiment. When Problem 2 is inconsistent, Qui et al. [32], Lei et al. [18], and Peng [31] established iterative methods over
the (skew–)symmetric, and (skew–)symmetric P –commuting matrices. On the other
hand, Liao et al. [20], Huang et al. [13] and Zhao et al. [38] derived an explicit
expressions of the least squares solution to Problem 2 when X is (skew–)symmetric
ON THE MATRIX NEARNESS PROBLEM FOR...
637
and .P; Q/–orthogonal symmetric matrix, respectively. Moreover, Sarduvan et al.
established the explicit forms of the best approximate solutions of Problem 2 when
X is .P; Q/–orthogonal (skew–)symmetric matrix [33].
And now, consider the following two problems.
Problem 3. For given matrices A1 2 Rm1 ;n , B1 2 Rn;p1 , C1 2 Rm1 ;p1 , A2 2
Rm2 ;n , B2 2 Rn;p2 , and C2 2 Rm2 ;p2 , find XO 2 ˝ such that
ŒA1 XO B1
C1 ; A2 XO B2
C2
H
D min kŒA1 XB1
X2˝
C1 ; A2 XB2
C2 kH
where ˝ is anyone of the sets of special matrices such as symmetric, skew–symmetric,
Hermitian, reflexive etc.
Problem 4. Let SE2 be the solution set of Problem 3. For a given matrix X0 2 Rn ,
find XO 2 SE2 such that
XO
X0 D min kX
X2SE2
X0 k :
Research on solving a pair of matrix equations has been actively ongoing for past
years. For instance, Mitra [24] and Navarra [25] established conditions for the existence of a solution and a representation of a general common solution to Problem
3. Also, Özgüler et al. [26], Woude [35], Wang [37], and Liu [22] derived necessary
and sufficient conditions for existence of common solution to Problem 3. Moreover,
Dehghan et al. [8] obtained conditions for the existence of .R; S/–(skew–)symmetric
solution of Problem 3. Ding et al. [9] presented an iterative method for solving a pair
of inconsistent matrix equations.
Besides the works on finding the conditions for the existence of common solution
to Problem 3, there are some valuable efforts on solving of the matrix nearness problem for a pair of matrix equations. For example, in the case that the matrix equations
in Problem 3 are consistent, iterative algorithms were presented for solving Problem 4 with certain constraints on solution such as symmetric, reflexive, bisymmetric,
generalized centro–symmetric, and generalized reflexive matrices in [3, 7, 29, 30, 34].
Cai et al. [2] and Chen et al. [4] derived iterative algorithms over bisymmetric and
symmetric solutions, respectively, in the case that the matrix equations in Problem 4
are inconsistent.
It is noteworthy that when the pair of matrix equations is inconsistent, its least
squares solutions with minimum norm cannot be obtained by GSVD and CCD. In
order to over come this difficulty, Liao, Lei [19] and Liao et al. [21] derived a different
approach based on the projection theorem. Therefore, they could used the method of
GSVD and CCD to obtain the solution.
In this paper, it is established the general expressions of the (skew–)symmetric
solutions to Problem 4 using kronecker product and Moore–Penrose inverse. Moreover, this general expressions are expanded for the matrix equations of the form
638
S. S.İMS.EK, M. SARDUVAN, AND H. ÖZDEMİR
.A1 XB1 ; : : : ; Ak XBk / D .C1 ; : : : ; Ck /. Furthermore, a comparative table depending
on some numerical examples in the literature is given.
2. P RELIMINARY R ESULTS
The vector x0 2 Rn;1 is a least squares solution (LSS) to the inconsistent system
of linear equations Ax D g, where A 2 Rm;n , if and only if
.Ax
g/T .Ax
g/ .Ax0
g/T .Ax0
g/
for all x 2 Rn;1 [10].
The vector x0 2 Rn;1 is the best approximate solution (BAS) to the inconsistent
system of linear equations Ax D g, where A 2 Rm;n , if and only if
(1) .Ax g/T .Ax g/ .Ax0 g/T .Ax0 g/ for all x 2 Rn;1 ,
(2) x T x > x0 T x0 for all x 2 Rn;1 n fx0 g satisfying .Ax g/T .Ax g/ D
.Ax0 g/T .Ax0 g/ [10].
It is noteworthy that there may be many LSS for an inconsistent system of linear
equations. In addition, a LSS may not be the BAS while the BAS is always a LSS.
However, the BAS is always unique.
If it is assumed that the matrix equation AXB D C , where A 2 Rm;n , B 2 Rp;r ,
C 2 Rm;r are known nonzero matrices and X 2 Rn;p is the matrix of unknowns,
is inconsistent, then it may be asked to find a matrix X such that kAXB C k is
minimum. A matrix satisfying this condition is called an approximate solution to
the matrix equation. The matrix XO 2 Rn;p is defined to be the BAS to the matrix
equation AXB D C if and only if
(1) kAXB C k AXO B C for all X 2 Rn;p ,
n o
(2) kXk > XO for all X 2 Rn;p n XO satisfying kAXB C k D AXO B C .
We note that a vector k 2 Rmn;1 will stand for the vector vec.K/ in the rest of the
text, where K 2 Rm;n .
It is known that the matrix equation AXB D C can be equivalently written as
B T ˝ A x D c:
(2.1)
Consequently, the solutions of a matrix equation AXB D C can be obtained by considering the usual system of linear equations (2.1) instead of the matrix equation
AXB D C . Now, we will give the following Lemma which can be proved easily.
Lemma 1 ([10]). Suppose that Sg is the set of all solutions to the consistent system
of linear equations Ax D g, where A 2 Rm;n is a known matrix, g 2 Rm;1 is a known
vector, and x 2 Rn;1 is the vector of unknowns. For a given vector x0 2 Rn;1 , the
vector xO 2 Sg satisfying
kxO
x0 k D min kx
x2Sg
x0 k
ON THE MATRIX NEARNESS PROBLEM FOR...
639
is given by
xO D A g C I
A A x0 :
Lemma 2. Let Se be the set of all least squares solutions to the system of linear
equations Ax D g which do not need to be consistent, where A 2 Rm;n is a known
matrix, g 2 Rm;1 is a known vector, and x 2 Rn;1 is the vector of unknowns. For a
given vector x0 2 Rn;1 , the vector xO 2 Se satisfying
kxO
x0 k D min kx
x2Se
x0 k
is given by
xO D A g C I
A A x0 :
Proof. If the system is consistent, then the proof is clear from Lemma 1. Now, let
the system be inconsistent. Then, the normal equations of the system is
AT Ax D AT g
(2.2)
which is consistent. So, from Lemma 1, the BAS of the inconsistent system Ax D g
is
xO D .AT A/ AT g C I .AT A/ .AT A/ x0
or, equivalently, from [10, Theorem 6.2.16]
xO D A g C I
A A x0 :
It is noteworthy that the structures of xO in Lemmas 1 and 2 are exactly the same.
Remark 1. The minimization problem
min kX
X0 k
is equivalent to the minimization problem
1
min X
X0 C X0T
2
over the set of RnS since
2
2
1
1
X0 C X0T
X0 X0T
C
; 8X 2 RnS :
2
2
So, the matrix 12 X0 C X0T instead of the matrix X0 is taken to find the symmetric
solutions of Problems 4 if the matrix X0 is not symmetric.
Similarly if the matrix X0 is not skew–symmetric, then the matrix 21 X0 X0T
instead of the matrix X0 is taken to find the skew–symmetric solutions of Problem 4.
kX
X0 k 2 D X
640
S. S.İMS.EK, M. SARDUVAN, AND H. ÖZDEMİR
3. T HE (S KEW–)S YMMETRIC S OLUTION OF P ROBLEM 4
Our aim is to find a symmetric solution of Problem 4 with an arbitrary matrix
X0 2 Rn . To do this, let us consider the quartet of matrix equations
A1 XB1 D C1
B1T X AT1 D C1T
A2 XB2 D C2
B2T X AT2 D C2T
or, equivalently, the usual system of linear equations
2
3
2
B1T ˝ A1
vec .C1 /
6 A1 ˝ B T 7
6 vec C T
1 7
1
6
6
4 B T ˝ A2 5 x D 4 vec .C2 /
2
vec C2T
A2 ˝ B2T
3
7
7:
5
(3.1)
In view of Lemma 2, the solution vector of the matrix nearness problem of the system
(3.1) is
3
2
3
2
3 2
3 2
B1T ˝ A1
vec .C1 /
B1T ˝ A1
B1T ˝ A1
T 7 6
T 7
6 A1 ˝ B T 7 6 vec C T 7
6
1 7 6
1
7 C x0 6 A1 ˝ B1 7 6 A1 ˝ B1 7 x0 ;
xO D 6
T
T
T
4 B ˝ A2 5 4 vec .C2 / 5
4 B ˝ A2 5 4 B ˝ A2 5
2
2
2
T
T
vec C2
A2 ˝ B2T
A2 ˝ B2T
A2 ˝ B2
(3.2)
where x0 D vec. 21 .X0 C X0T //.
Thus, we have the following theorem within the framework of those.
Theorem 1. Let A1 2 Rm1 ;n , B1 2 Rn;p1 , C1 2 Rm1 ;p1 , A2 2 Rm2 ;n , B2 2
Rn;p2 , C2 2 Rm2 ;p2 , X0 2 Rn are known matrices, and x0 D vec. 21 .X0 C X0T //.
Then the symmetric solution XO 2 RnS of Problem 4 is given as in (3.2) in view of
xO D vec.XO /.
If it is required to find skew–symmetric solution of Problem 4, then vec CiT is
taken instead of vec CiT , i D 1; 2.
By continuing with the same idea, Theorem 1 can be extended to k matrix equations where k is an arbitrary positive integer.
Theorem 2. Let Ai 2 Rmi ;n , Bi 2 Rn;pi , Ci 2 Rmi ;pi , i D 1; 2; : : : ; k, are known
matrices and
SE D fX j X 2 ˝; kŒA1 XB1
C1 ; : : : ; Ak XBk
Ck kH D mi ng:
For a given matrix X0 2 Rn the symmetric solution XO 2 SE satisfying
XO
X0 D min kX
X 2SE
X0 k
(3.3)
ON THE MATRIX NEARNESS PROBLEM FOR...
is given by
2
B1T ˝ A1
6 A1 ˝ B T
1
6
6
::
xO D 6
:
6
4 B T ˝ Ak
k
Ak ˝ BkT
3 2
7
7
7
7
7
5
vec .C1 /
vec C1T
::
:
6
6
6
6
6
4 vec .Ck /
vec CkT
3
7
7
7
7 C x0
7
5
2
B1T ˝ A1
A1 ˝ B1T
::
:
6
6
6
6
6
4 B T ˝ Ak
k
Ak ˝ BkT
641
3 2
7
7
7
7
7
5
B1T ˝ A1
A1 ˝ B1T
::
:
6
6
6
6
6
4 B T ˝ Ak
k
Ak ˝ BkT
3
7
7
7
7 x0 ;
7
5
(3.4)
where x0 D vec. 21 .X0 C X0T // and, xO D vec.XO /.
Similarly, vec CiT is taken instead of vec CiT , i D 1; 2; : : : ; k, for finding
skew–symmetric solution.
TABLE 1. A comparative table for the examples chosen from the literature
" (F)
kX X0 k
43.6600
kŒA1 XB1
C1 ; A2 XB2
4.5687e+003
C2 kH
1
43.6600
0.4366
4.5687e+003
45.6873
0.4366
0.0044
45.6873
0.4569
0.0044
4.3660e-005
0.4569
0.0046
4.3660e-005
28.7113
0.0046
6.1988e+003
33.6729
0.2871
4.5687e+003
66.9884
0.3367
0.0029
48.1408
0.6199
0.0034
2.8711e-005
0.4568
0.0062
3.3673e-005
413.4852
0.0046
2.6688e-021
413.4852
69.9995
4.4970e-021
9.0727e-023
69.9995
1.3436e-021
0.01
Example 2
in [19]
0.0001
0.000001
1
0.01
Example 3
in [21]
0.0001
0.000001
Example 1 in [29]
Example 4.1 in [34]
(F) " is as given in [19, 21].
We close this section with a comparative table (Table 1) consisting examples
chosen from the literature. In each cell, the first value is the result obtained by the
642
S. S.İMS.EK, M. SARDUVAN, AND H. ÖZDEMİR
method proposed in this work while the second one is the result in the referenced
work. All the computations have been performed using Matlab 7.5.
4. C ONCLUSIONS
To solve matrix equations system problems become relatively difficult when it is
used matrix decompositions. For example, if the matrix equations are inconsistent,
the matrix decompositions GSVD and CCD can not be individually used to solve
them, and the difficulty lies in the fact that the invariance of the Frobenius norm
does not hold for general nonsingular matrices in these decompositions [19]. For this
reason, these kinds of Problems are usually solved using iterative methods. However,
it is a well known fact that solving these kinds of problems by elementary methods,
which are very simple and elegant, eliminates errors caused by processes of iteration.
Due to these kinds of facts, in our opinion, it is better to give the explicit analytical
expressions of the solutions obtained by elementary methods instead of giving, especially, the implicit solutions obtained by iterative methods for inconsistent matrix
equations encountered in most of physical problems.
If the dimensions and elements of the matrices included in the problems are large
and sparse, it is clear that the computing processes, especially in the elementary methods, have contained highly large number of terms. Therefore, elementary methods
may not be useful with the current computer technology in these kinds of situations.
On the other hand, the speed of technological developments is incredible. So, we
believe that these difficulties will be disappeared in the nearest future. Consequently,
within the framework of these considerations, to establish the solutions as in this note
are important not only mathematical point of view but also practically.
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Authors’ addresses
S. S.ims.ek
Kırklareli University, Department of Mathematics, TR39100, Kırklareli, Turkey
E-mail address: sinem.simsek@klu.edu.tr
M. Sarduvan
Sakarya University, Department of Mathematics, TR54187, Sakarya, Turkey
E-mail address: msarduvan@sakarya.edu.tr
ON THE MATRIX NEARNESS PROBLEM FOR...
H. Özdemir
Sakarya University, Department of Mathematics, TR54187, Sakarya, Turkey
E-mail address: hozdemir@sakarya.edu.tr
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