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PureMathematicsnØêÆ,2022,12(8),1360-1369
PublishedOnlineAugust2022inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2022.128149
Ý•§AXB=CÓ†4‰ê
•¦)
ùùùøøø)))§§§¸¸¸•••
∗
ʵŒÆêƆOŽ‰ÆÆ§2Àô€
ÂvFϵ2022c719F¶¹^Fϵ2022c819F¶uÙFϵ2022c830F
Á‡
Ì‚ÝkaÈ{¤¿…3¯õ‰Æ+•2•A^"Ý•§AXB=C3A½8Ü
a¦)Ú•z¯K3ó§+•k-‡A^"©ÏLÝKroneckerÈÚMoore-
Penrose2Â_Ý•§AXB=CkÓ†)¿‡^‡Ú)Lˆª"3vk Ó†)ž§
‰Ñ•§Ó†4‰ê•¦)"3Ø©"!§‰Ñ•§¦)ꊎ{†êŠ~f"
'…c
Ó†Ý§4‰ê)§•¦)§Moore-Penrose2Â_§KroneckerÈ
LeastSquaresCirculantSolution
oftheMatrixEqautionAXB=C
withtheLeastNorm
YuzheCao,ShifangYuan
∗
SchoolofMathematicsandComputationScience,WuyiUniversity,JiangmenGuangdong
Received:Jul.19
th
,2022;accepted:Aug.19
th
,2022;published:Aug.30
th
,2022
∗ÏÕŠö"
©ÙÚ^:ùø),¸•.Ý•§AXB=CÓ†4‰ê¦)[J].A^êÆ?Ð,2022,12(8):1360-1369.
DOI:10.12677/pm.2022.128149
ùø)§¸•
Abstract
Circulant matrices have been aroundfora longtimeand have been extensively usedin
many scientificareas.Theproblemofsolvingandminimizing thematrixAXB= Cin a
specific setclass hasimportant applications inengineering andotherrelatedfields.In
thispaper,by usingKroneckerproductandMoore-Penrose generalizedinverseofthe
matrices, thenecessaryand sufficientconditionsforAXB= Chaving circulant solution
areobtained.Wederivetheexpressionoftheleastsquarescirculantsolutionofthe
matrixequationAXB= Cwiththeleastnormwhenthereisnocirculantsolution.In
thelastsection,thenumericalalgorithmandnumericalexamplesarealsogiven.
Keywords
CirculantMatrix,LeastNormSolution,LeastSquaresSolution,Moore-PenroseIn-
verse, TheKroneckerProduct
Copyright
c
2022byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution InternationalLicense(CC BY4.0).
http://creativecommons.org/licenses/by/4.0/
1.Úó
‚5Ý•§¦)¯K9ƒ A•¦ ¯K´Cc5ꊓê+• ¥ïÄÚ?Ø-‡
‘Kƒ˜,§3(O,XÚ£O,gÄ››n+•¥kX2•A^.
Cc5,Ý•§
AXB= C(1)
9ƒ'•§3顧‡é¡§Vé¡õ«Ýa¥)ÚÙ•¦)ïÄØ?
Ð[1–8].‚5Ý•§•¦)˜„5`Ø´•˜ ,§4‰ê•¦)˜„5`´
•˜.
••B£ã¯K§©^C
m×n
L«m×nEêÝ8Ü.A
T
L«ÝA=˜,A
∗
L«Ý
AÝ=˜,A
+
L«ÝAMoore-Penrose2Â_,I
n
L«nü Ý.
DOI:10.12677/pm.2022.1281491361nØêÆ
ùø)§¸•
½Â1.1.c= (c
0
,c
1
,···,c
n−1
)
T
´n‘Eê•þ,¡XeÝ
C=











c
0
c
1
c
2
···c
n−1
c
n−1
c
0
c
1
···c
n−2
c
n−2
c
n−1
c
0
···c
n−3
···············
c
1
c
2
c
3
···c
0











(2)
´cÓ†Ý§¿P•
C= circ(c) = circ(c
0
,c
1
,···,c
n−1
).(3)
·‚^cir(n)L«NnÓ†Ý8Ü.Ó†ÝÄgÑy1846cCatalan˜ŸêÆØ©¥,†
Ó†Ý—ƒƒ'´ToeplitzÝ.8c Ó†Ý3—èÆ,Ôn9V-AÛ¥kX-‡A^.
'uÓ†Ý?˜Ú5Ÿ,ë„©z[9–11].
½Â1.2.A=(a
ij
)∈C
m×n
.Pa
i
=(a
1i
,a
2i
,···,a
mi
),i=1,2,···,n.rAÀŠ•þ.†
ŽfP•
vec(A) = (a
1
,a
2
,···,a
n
)
T
.(4)
é•þx= (x
1
,···,x
n
)
T
∈C
n×1
,kxk
2
=
p
P
n
i=1
|x
i
|
2
L«•þ2-‰ê.d •þ2-‰êpX
eÝFrobenius‰ê
kAk= kvec(A)k
2
.(5)
‰½Ý
A∈C
m×n
,B∈C
n×s
,C∈C
m×s
.(6)
©ò3Ó†Ýa¥•ÄÝ•§AXB= CXe¯Kµ
¯KI:¦Xe8Ü
T
s
=
n
X∈C
n×n
: AXB= C
o
.(7)
¯KII:e¯KI¥T
s
= ∅(˜8),¦Xe8Ü
T
l
=
n
X∈C
n×n
: ||AXB−C||= min
o
.(8)
¯KIII:e¯KI¥T
s
= ∅,¦
b
X∈T
l
¦
k
b
Xk
2
=min
X∈T
l
kXk
2
.(9)
©ò312!‰ÑÓ†ÝeZ5ŸÚ'uÝKroneckerÈÚMoore-Penrose2Â_A
KÚn.313!¥§·‚ò|^12!ƒ'(؉ÑÝ•§AXB=C'u±þn‡¯K¥)
DOI:10.12677/pm.2022.1281491362nØêÆ
ùø)§¸•
Lˆª.314!¥,ò‰Ñ¦)ꊎ{†êŠ~f5`²·‚Ž{Œ15.
2.Ó†Ý5ŸÚAKÚn
d½Â1.1•˜†Ý
P=





010···0
001···0
000···0
.
.
.
.
.
.
.
.
.
.
.
.
100···0





= circ(p),(10)
Ù¥p=(0,1,0,···,0)
T
.P´AÏÓ†Ý§·‚¡ƒ•£ Ý.dÝ3©¥kX-‡
Š^.
·‚ò|^P9•þc5L«Ó†ÝC.
·K2.1.••BO,5½P
0
= I
n
.C= cir c(c) = circ(c
0
,c
1
,···,c
n−1
).K
C= circ(c) =
n−1
X
k=0
c
k
P
k
.(11)
y²w,·‚k
P
2
= circ(0,0,1,0,···,0),P
3
= circ(0,0,0,1,0,···,0),···,P
n
= circ(1,0,···,0) = I
n
.(12)
d½Â1.1•C= circ(c) =
P
n−1
k=0
c
k
P
k
.
·K2.2.A= circ(a) = circ(a
0
,a
1
,···,a
n−1
)ÚB= circ(b) = circ(b
0
,b
1
,···,b
n−1
).KAB´Ó
†Ý…
AB= BA= C= circ(c
0
,c
1
,···,c
n−1
),(13)
Ù¥
c
k
=
X
(i,j)∈S
k
a
i
b
j
,k= 0,1,···,n−1,S
k
= {(i,j) : i+j= kmodn}.(14)
y²d·K2.1•A=
P
n−1
i=0
a
i
P
i
=a
0
P
0
+a
1
P
1
+···+a
n−1
P
n−1
ÚB=
P
n−1
j=0
b
j
P
j
=
b
0
P
0
+b
1
P
1
+···+b
n−1
P
n−1
•AB= BA¿…
C= AB= (a
0
P
0
+a
1
P
1
+···+a
n−1
P
n−1
)(b
0
P
0
+b
1
P
1
+···+b
n−1
P
n−1
) =
n−1
X
k=0
c
k
P
k
,
Ù¥
c
k
= a
0
b
k
+a
1
b
k−1
+···+a
n−1
b
k+1
,k= 0,1,···,n−1.
dª(12)•
c
k
=
X
(i,j)∈S
k
a
i
b
j
,k= 0,1,···,n−1,
DOI:10.12677/pm.2022.1281491363nØêÆ
ùø)§¸•
Ù¥S
k
= {(i,j) : i+j= kmodn}.
üÓ†ÝÚ´Ó†Ý,˜‡Eꦘ‡Ó†ÝE´˜‡Ó†Ý.dd•,l•þÝ
5wÓ†Ý8Ü´˜‡‚5˜m.
½Â2.1.én≥2,P
M
n
=

vec(P
0
),vec(P
1
),vec(P
2
),···,vec(P
n−1
)

∈R
n
2
×n
.(15)
Ún2.1.C= circ(c) = circ(c
0
,c
1
,···,c
n−1
).K
vec(C) = M
n
c.(16)
y²dC= circ(c) =
P
n−1
k=0
c
k
P
k
= c
0
P
0
+c
1
P
1
+···+c
n−1
P
n−1
Œ
vec(C) = vec(P
0
)c
0
+vec(P
1
)c
1
+···+vec(P
n−1
)c
n−1
= M
n
c.
Ún2.1•xÓ†Ý8Ü´˜‡‚5˜mA,•·‚ïÄÓ†ÝJøóä.
Ý•§AXB=C3Ó†Ýa),Ù¢Ÿ´¦)¤X•þx.•d,·‚òrÝ•
§AXB= C=z•·‚š~ÙG‚5•§.•=zÚ¦)ƒA•§,·‚I‡e¡½Â9Ún.
Ún2.2.[12]A∈C
m×n
,b∈C
m
,‚5•§Ax= bk)¿©7‡^‡´AA
+
b= b.k)ž•
§Ï)•
x= A
+
b+(I
n
−A
+
A)z,∀z∈C
n
.
Ún2.3.[12]A∈C
m×n
,b∈C
m
,K؃N‚5•§Ax= b•¦){x∈C
n
: kAx−bk
2
=
min}•
x= A
+
b+(I
n
−A
+
A)z,∀z∈C
n
.
½Â2.2.A= (a
ij
) ∈C
m×n
,B= (b
ij
) ∈C
p×q
.¡
A⊗B=




a
11
Ba
12
B···a
1n
B
a
21
Ba
22
B···a
2n
B
············
a
m1
Ba
m2
B···a
mn
B




∈C
mp×nq
(17)
•AÚBKroneckerÈ.
Ún2.4.[12]éÝA∈C
m×n
,X∈C
n×n
,B∈C
n×s
k
vec(AXB) = (B
T
⊗A)vec(X).(18)
3.̇(Ø
e¡·‚‰ÑÝ•§AXB= C3Ó†Ýak)¿‡^‡,¿‰Ñ)Lˆª.
DOI:10.12677/pm.2022.1281491364nØêÆ
ùø)§¸•
½n3.1.‰½ÝA∈C
m×n
,B∈C
n×s
,C∈C
m×s
.-
R= (B
T
⊗A)M
n
.(19)
P
X= circ(x) = circ(x
0
,x
1
,···,x
n−1
).
KÝ•§AXB= Ck)¿©7‡^‡´
RR
+
vec(C) = vec(C).(20)
•§k)ž,•§Ï)•
T
s
=
n
X= circ(x) : x= R
+
vec(C)+

I
n
−R
+
R

z,∀z∈C
n
o
.(21)
y²eAXB= C,Kkvec(AXB) = vec(C).dÚn2.4Ú2.1•
(B
T
⊗A)M
n
x= vec(C).
=
Rx= vec(C).(22)
Ý•§AXB= Cdu•§(22).dÚn2.2Ý•§AXB= Ck)¿©7‡^‡´
RR
+
vec(C) = vec(C).
•§k)ž,ÙÏ)•
x= R
+
vec(C)+

I
n
−R
+
R

z,∀z∈C
n
.
Ý•§AXB= C3Ó†Ýa¥Ã)ž,·‚ŒdXe½n¦Ù•¦).
½n3.2.^‡ÚÎÒÓ½n3.1.RR
+
vec(C)6=vec(C)ž,Ý•§AXB=C3Ó†Ýa
•¦)8T
l
ŒL«•
T
l
=
n
X= circ(x) : x= R
+
vec(C)+

I
n
−R
+
R

z,∀z∈C
n
o
.(23)
y²dÚn2.1Ú2.4•
||AXB−C||=||vec(AXB)−vec(C)||
2
=||(B
T
⊗A)vec(X)−vec(C)||
2
=||(B
T
⊗A)M
n
x−vec(C)||
2
=||Rx−vec(C)||
2
.
DOI:10.12677/pm.2022.1281491365nØêÆ
ùø)§¸•
dÚn2.3•µkAXB−Ck= min…=
x= R
+
vec(C)+

I
n
−R
+
R

z,∀z∈C
n
.(24)
Ý•§AXB=C3Ó†Ýa ¥Ã)ž,·‚ŒdXe½n3þ¡‰Ñ•¦)8
éäk4‰ê).
½n3.3.^‡ÚÎÒÓ½n3.2,¿-
K= I
n
−R
+
R.(25)
KÝ•§AXB= C•3•˜4‰ê•¦)
b
X∈T
l
.P
b
X= circ(ˆx).
K
b
XŒL«•
vec(
b
X) = M
n
R
+
vec(C)−M
n
K(M
n
K)
+
M
n
R
+
vec(C).(26)
y²dÚn2.1•
vec(X) = M
n
x.
d½n3.2•
k
b
Xk
2
=min
X∈T
l
kXk
2
=min
X∈T
l



vec(X)



2
2
=min
z



M
n
(I
n
−R
+
R)z−(−M
n
R
+
vec(C))



2
2
=min
z



M
n
Kz−(−M
n
R
+
vec(C))



2
2
.
dÚn2.3•
min
z



M
n
Kz−(−M
n
R
+
vec(C))



2
2
)•
z= −(M
n
K)
+
M
n
R
+
vec(C)+[I
n
−(M
n
K)
+
M
n
K]u,∀u∈C
n
.(27)
òþª“\ª(24)
ˆx= R
+
vec(C)−K(M
n
K)
+
M
n
R
+
vec(C)+K(I
n
−(M
n
K)
+
M
n
K)u,∀u∈C
n
.(28)
k
vec(
b
X) = M
n
ˆx= M
n
R
+
vec(C)−M
n
K(M
n
K)
+
M
n
R
+
vec(C).
DOI:10.12677/pm.2022.1281491366nØêÆ
ùø)§¸•
4.ꊎ{ÚêŠ~f
Šâ½n3.1Ú½n3.2,3.3,y‰Ñ¯KI¯KIIIXeŽ{.
Ž{:
(1)Ñ\A∈C
m×n
,B∈C
n×s
,C∈C
m×s
9M
n
;
(2)OŽÑR= (B
T
⊗A)M
n
;
(3)eRR
+
Vec(C) = Vec(C)¤á§K¯KIk);OŽÑ
K= I
n
−R
+
R
Ú
x= R
+
Vec(C)+Kz,∀z∈C
n
,
Œ¯KIÏ)•
X= circ(x);
(4)eRR
+
Vec(C) 6= Vec(C)§K¦¯KII).OŽÑ
x= R
+
Vec(C)+Kz,∀z∈C
n
,
Œ¯KIIÏ)•
X= circ(x);
(5)OŽÑ
ˆx= R
+
vec(C)−K(M
n
K)
+
M
n
R
+
vec(C),
¯KIII•˜)
b
X= circ(ˆx).
~4.1.
A=






23−6
854
713
1211−15






,B=




45
−78
12




9
C
1
=






219−49
71119
6632
705−91






,C
2
=






140−89i−65+60i
2−50i3+250i
−10−50i160+180i
400−300i−5+200i






.
DOI:10.12677/pm.2022.1281491367nØêÆ
ùø)§¸•
(1)C= C
1
ž§dOŽ•
R=

















−19−3556
1−350
24−36−10
−44−133161
22−8−29
889473
497343
11843−41

















,
RR
+
Vec(C) = Vec(C),K= I
3
−R
+
R= 0…
R
+
Vec(E) = (1,−2,3)
T
.
¯KI)•
X= circ(1,−2,3).
(2)C= C
2
ž§RR
+
Vec(C) 6= Vec(C)…
R
+
Vec(C) = x,
Ù¥
x= (0.4595+1.0952i,−0.7605+1.7234i,1.8766−0.1500i).
Ï•K= 0,¯KIIÚ¯KIII)ƒÓ,Ù•
X= circ(0.4595+1.0952i,−0.7605+1.7234i,1.8766−0.1500i).
Ä7‘8
2ÀŽpg ,‰ÆÄ7-:‘8(2019KZDXM025),ʵŒÆleéÜïuÄ7]Ï‘8(
2019WGALH20).
ë•©z
[1]Hu,X.Y.andDeng,Y.B.(2003)OntheSolutionsOptimalApproximationoftheEquation
A
T
XB= CoverASR
m×m
.NumericalMathematics,5,59-62.
[2]Liao, A.P., Bai, Z.Z.andLei, Y.(2006)BestApproximate SolutionofMatrixEquationAXB+
CXD= E.SIAMJournalonMatrixAnalysisandApplications,27,675-688.
https://doi.org/10.1137/040615791
DOI:10.12677/pm.2022.1281491368nØêÆ
ùø)§¸•
[3]¸•,S²,X.Ý•§AXB+CYD=Eé¡4‰ê•¦)[J].OŽêÆ,
2007,29(2):203-216.
[4]Yuan,S. andLiao,A.(2014)LeastSquaresHermitianSolution oftheComplexMatrixEqua-
tion AXB +CXD =E with theLeast Norm. JournaloftheFranklinInstitute, 351, 4978-4997.
https://doi.org/10.1016/j.jfranklin.2014.08.003
[5]Liu,Z.,Zhou,Y.,Zhang,Y.,etal.(2019)SomeRemarksonJacobiandGauss-Seidel-Type
Iteration Methods for the Matrix Equation AXB = C. AppliedMathematicsandComputation,
354,305-307.https://doi.org/10.1016/j.amc.2019.02.014
[6]Dehghan, M.and Shirilord,A. (2019)AGeneralizedModifiedHermitianand Skew-Hermitian
Splitting(GMHSS)MethodforSolvingComplexSylvesterMatrixEquation.AppliedMathe-
maticsandComputation,348,632-651.https://doi.org/10.1016/j.amc.2018.11.064
[7]Wu,N.C.,Liu,C.Z.and Zuo,Q. (2022)On theKaczmarz MethodsBasedon RelaxedGreedy
Selection for SolvingMatrix EquationAXB =C. JournalofComputationalandAppliedMath-
ematics,413,ArticleID:114374.https://doi.org/10.1016/j.cam.2022.114374
[8]Safarzadeh,M.,SadeghiGoughery,H.andSalemi,A.(2022)Global-DGMRESMethodfor
MatrixEquationAXB=C.InternationalJournalofComputerMathematics,99,1005-1021.
https://doi.org/10.1080/00207160.2021.1942459
[9]•µû,••Ÿ.AÏÝ[M].®:˜uŒÆч,2001.
[10]Davis,P.J.(1994)CirculantMatrices.AMSChelseaPublishing,NewYork.
[11]Arup,B.andKoushik,S.(2018)RandomCirculantMatrices.CRCPress,BocaRaton,FL.
[12]•u.ÝØ[M].®:‰Eч,2001.
DOI:10.12677/pm.2022.1281491369nØêÆ

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