2 - Computational Mathematics

 2010 9
Sep., 2010
Journal On Numerical Methods and Computer Applications
31
3
Vol.31, No.3
Æ
Æ
Æ
!
*1)
(
,
510640;
,
(
523808)
,
(
,
(
510640)
,
100192)
510640)
. , , . , , Moor-Penrose , : ; ; ; MR (2000) !"#: 65F18, 15A09
,
,
Kronecker
n
.
"
THE INVERSE GENERALIZED EIGENVALUE PROBLEM
AND THE OPTIMAL APPROXIMATION FOR
HERMITIAN-REFLEXIVE MATRICES
Wang Jiangtao
(Department of Mathematics, South China University of Technology, Guangzhou 510640, China;
School of Business and administration, South China University of Technology,
Guangzhou 510640, China)
Zhang Zhongzhi
(School of Computer Science and Technology, Dongguan University of Technology,
Dongguan 523808, Guangdong, China)
Xie Dongxiu
(School of Sciences , Beijing Information Science and Technology University, Beijing 100192, China)
Lei Xiuren
(Department of Mathematics, South China University of Technology, Guangzhou 510640, China)
*
1)
2010
! "#
1
:
2
.
(10971058),
Æ$%&! (61N0810810).
3
233
Abstract
The best approximation problem with the given spectral constraints is usually used to
correct a stiffness and a mass matrix in the Vibration Control. Based on the denotative theorem of Hermitian-reflexive matrices, the author discuses the inverse generalized eigenvalue
problem of Hermitian-reflexive and obtain the the general expression of the solution by using
the straightened matrices and the Kronecker product of matrices. Furthermore, for any given
complex matrices of dimension n, the expression of solution for its optimal approximate are
presented by using the Moore-Penrose generalized inverse and the best approximation theory.
Keywords: Hermitian-Reflexive Matrices; Generalized eigenvalue; Straightened matrix; Optimal approximation
2000 Mathematics Subject Classification: 65F18, 15A09
#$
1.
$%'&, '()Æ !"!#*(+!$)!#, ,$#*%*-+,&.!.'/0/("1.'/0)* (0 [1]). 1+, ,2#$!%!"&%&-.!- 2
)*.
ω ,ω ,··· ,ω
1
2
m (m
≤ n)
(m
3./3, φ1, φ2 , · · · , φm
φ = (φ1 , φ2 , · · · , φm ),
(04!%+. 4
2
Ω2 = diag(ω12 , · · · , ωm
).
K &'-2!%&-., M &'-2!55-., 1676+,!7238:
(1) ' /(): Kφ = M φΩ ; (2) *+8: K = K, M = M.
Æ976%47238! K, M ,:9Æ9;:*+,-!.9 K, M !"1.'/0
)*.
5<6;/0, "'12!78, /(=<:9=,(3*-45%&-. K !6:7
'0 K >55-. M !6 : 0 M . 8? M.Baruch , ; (0 [2]) < $!@ = ÆA >? B
C9((3, *--2 K > M , D
Æ-. K, M @A76
2
T
0
T
0
0
0
min(K
M ) − (K0
M0 ),
A& K > M (7638 (1) > (2) !9.
,:E+FB;,-!%&12>5512, @,-;:(Æ!<: / 3>0 4!<:
%+,Æ)**+,&Æ9-.!"1.'/0)*>> A0.!-.=G)*.
?9"1.'/0)*$>*?@*?A#CB?3H@,IA:B"C!4<, CD
DDJEE*F!.G. .9.'/0)*>"1.'/0)*!HKIE4JÆ!LFM,
0 FG [3-9]. 1 +, FG [3] =<-.! QR 19$L7J! H N , HKJ I *+-."1
.'/0)*, 45)*9!ÆG:HJ; FG [5] =<KO!(3PÆ9"1'/0I)*;
FG [7] K<LJ(3Æ9"1.'/0)*; FG [8] &=<-.!ML019>-.1Q
3HKJ*+2R*+-.!"1 '/0I)*. *-..'/0)*>"1.'/0 )*
!HK, KE(L<-.! QR 19?ML019,(3, MFS=<-.TM,(3PNU
NOV'3I-.!"1.'/0)*.
WONWOXO
2010
PQ, DRÆSMF&@<!PY>QT. 4 C :U m × n +R-.Z[, R :
U m×n +I-.Z[,I :U k \SV-., SR , ASR 1B:U n \*+-.>I*
+-.!Z[. HC , AHC 1B:U n \NOV'-.>INOV'-.Z[,U C
:U n \P-.Z[. A , A 1B:U A ! Moore-Penrose "1.>TQRS, tr(A) :U
A !], *9-. A, B ∈ C
, A⊗B ∈ C
:U A > B ! Kronecker ^, < A, B > :
U A > B !_^, A;1& < A, B >= tr(B A), ?U_^TV!W
#&
234
m×n
m×n
n×n
k
n×n
n×n
n×n
+
n×n
H
m×n
mp×nq
H
A =
< A, A > =
tr(AH A).
W., %4W#( Frobenius W#, CU C -FÆ XX!_^`a. · :UY5
! 2 W#.GRC :U n \"1IZ-.UF!Z[, D
n×n
2
n×n
V
%
A = (a
1.
GRC n×n = {P |P H = P, P 2 = In , P ∈ C n×n }.
ij )m×n ,
b ai = (a1i , a2i, · · · , ami)T
(i = 1, 2, · · · , n).
4
vec(A) = (aT1 , aT2 , · · · , aTn )T .
+Y5 vec(A) &-. A !TM (YL).
+M: AT = A, b a1 = (a11, a21, · · · , an1)T , a2 = (a22, a32, · · · , an2)T , · · · , an−1 =
(a(n−1)(n−1) , an(n−1) )T , an = ann . Kb
vecS (A) = (aT1 , aT2 , · · · , aTn−1 , aTn )T ∈ R
&
n(n+1)
2
K?-.TM!;1>FG [10] &!(3, WX4,2D).
Y 1. * X ∈ Rn×n, K,2cJFd
.
X ∈ SRn×n ⇐⇒ vec(X) = Kn vecS (X),
(1)
A& vec (X) ∈ R
S
⎛
⎜
⎜
⎜
⎜
⎜
Kn = ⎜
⎜
⎜
⎜
⎝
n(n+1)
2
, Kn
?,J#$,ei &SV-. In !Z i L,
e1
e2
e3
···
en−1
en
0
0
···
0
0
···
0
0
0
e1
0
···
0
0
e2
e3
···
en−1
en
···
0
0
0
..
.
0
0
..
.
0
e1
..
.
0
···
0
..
.
0
0
..
.
0
e2
..
.
0
···
..
.
···
0
..
.
e2
0
..
.
0
···
···
···
0
..
.
e1
···
···
0
..
.
en−1
0
..
.
en
0
0
0
···
0
e1
0
0
···
0
e2
···
0
en−1
0
⎞
⎟
0 ⎟
⎟
0 ⎟
⎟
.
.. ⎟
. ⎟
⎟
⎟
0 ⎠
en
(1.1)
(2)
X ∈ ASRn×n ⇐⇒ vec(X) = Tn vecS (X),
vecS (X), ei
, Tn
,
⎛
0 e2
e3 · · · en−1 en 0
0
···
⎜
0
···
0
0
0 e3 · · ·
⎜ 0 −e1
⎜
⎜ 0
0
0
0 −e2 · · ·
0
−e1 · · ·
⎜
Tn = ⎜ .
.
.
.
.
..
..
..
..
..
..
..
⎜ ..
···
.
.
.
⎜
⎜
⎝ 0
0
0
· · · −e1
0
0
0
···
A&
K%
0
0
0
?,J#$
···
0
−e1
0
0
···
0
0
···
0
0
en−1
0
..
.
en
0
..
.
···
···
···
0
0
..
.
0
0
..
.
−e2
0
···
0
en
0
−e2
···
0 −en−1
0
⎞
⎟
⎟
⎟
⎟
⎟
⎟.
⎟
⎟
⎟
0 ⎠
0
0
..
.
0
(1.2)
3
V
%
235
2.
#; P ∈ GRC n×n, + A ∈ C n×n &.9 P !NOV'3I-., +M176:
AH = A, P AP = A.
b[: n \NOV'3I-.!Z[& HCrn×n (P ) ( r :U3I-.), D
HCrn×n (P ) = {A|P AP = A, AH = A, A ∈ C n×n }.
W., -.Z[ HC (P ) > P !ZE:., MF[e;-. P (<;[\!.
9(, NOV'3I-."1.'/0)*f0.=G)* !#*g4+,:
n×n
r
\]. #; X ∈ C n×m , Λ = diag(λ1, λ2 , · · · , λm ) ∈ C m×m, 4
SA,B = (A, B)|AX = BXΛ, A, B ∈ HCrn×n (P ) .
(1.3)
*#;!-. A∗ , B∗ ∈ C n×n , Æ A, B ∈ SA,B @4
(A∗
B)
=
B ∗ ) − (A
min
(A,B)∈SA,B
(A∗
B ∗ ) − (A
B).
(1.4)
MF,-+,: $Z 2 h, XE)*9!]$8>\Æ8, ^]VA\Æ94:HJ; Z 3
h, #$Æ9)*!#0^3-f#0^1.
2.
'()
^_`ab c d c
4
P1 =
1
1
(I + P ), P2 = (I − P ),
2
2
[iXE P1 , P2 j&2R_e-., D`: P1 + P2 = I, P1P2 = 0. CU, ]$SVL2
R-. Q1 ∈ C n×r , Q2 ∈ C n×(n−r), @4 P1 = Q1QH1 , P2 = Q2QH2 , a4 Q = (Q1, Q2) ∈ C n×n,
K Q ∈ U C n×n.
Y 2[11] . 4
&
S=
&
M1
A|A = Q
0
0
M2
H
Q , M1 ∈ HC
r×r
, M2 ∈ HC
9(: S = HCrn×n(P ).
Y 3[12] . A ∈ C m×n, b ∈ C m ,
(1) b 8()U Ax = b :9!_( 3 8 (
Uc, 59&
(2)
AA+ b = b.
x = A+ b + (I − A+ A)y,
∀y ∈ C n .
x = A+ b + (I − A+ A)y,
∀y ∈ C n .
[0db8()U Ax = b !f=`a9&
(n−r)×(n−r)
.
WONWOXO
236
2010
*eg! M1, N1 ∈ HC r×r , M2, N2 ∈ HC (n−r)×(n−r), :,JFd
M1 = M11 + iM12 ,
N1 = N11 + iN12 ,
(2.1)
M2 = M21 + iM22 ,
N2 = N21 + iN22 .
(2.2)
A& M
∈ SRr×r , M12 , N12 ∈ ASRr×r , M21 , N21 ∈ SR(n−r)×(n−r), M22 , N22 ∈
ASR
.
11 , N11
(n−r)×(n−r)
?D) 1 ?D) 2 -fFG [8] !fg, [iXE,2;).
VY 1. #; X ∈ C n×m, Λ = diag(λ1, λ2, · · · , λm) ∈ C m×m, `4
H
Q X=
X1
X2
,
(2.3)
R1 = (X1T ⊗ Ir )Kr , R2 = (X1T ⊗ Ir )iTr , R3 = ((X1 Λ)T ⊗ Ir )Kr , R4 = ((X1 Λ)T ⊗ Ir )iTr ,
W1 = diag(Kr Tr Kr Tr ),
(2.4)
G1 = (R1 R2 −R3 −R4 ),
R5 = (X2T ⊗ In−r )Kn−r , R6 = (X2 ⊗ In−r )iTn−r , R7 = ((X2 Λ)T ⊗ In−r )Kn−r , R8 = ((X2 Λ)T ⊗
In−r )iTn−r , G2 = (R5 R6 −R7 −R8 ), W2 = diag(Kn−r Tn−r Kn−r Tn−r ). (2.5)
KZ[ SA,B b`, `A=h*:U&:
M1
A=Q
0
0
QH ,
M2
N1
B=Q
0
0
N2
QH ,
(2.6)
,C Mi, Ni , i = 1, 2 + (2.1), (2.2) J;1, `:
⎛
⎞
vec(M11 )
⎜
⎟
⎜vec(M12 )⎟
+
⎜
⎟
⎜ vec(N ) ⎟ = W1 (I − G1 G1 )x,
11 ⎠
⎝
vec(N12 )
(2.7)
⎛
⎞
vec(M21 )
⎜
⎟
⎜vec(M22 )⎟
+
⎜
⎟
⎜ vec(N ) ⎟ = W2 (I − G2 G2 )y,
21 ⎠
⎝
vec(N22 )
(2.8)
A& x ∈ C , y ∈ C
&egY5.
,2, XE)*9!]$8>iÆ8c^]Vfk=G9!:HJ.
VY 2. *#;-. A , B ∈ C , A1PY>;) 1 0K c`4
2r(r+1)
2(n−r)(n−r+1)
∗
∗
∗
∗
QH
1 A Q1 = A1 + iA2 ,
∗
∗
∗
QH
2 B Q2 = B3 + iB4 ,
∗
n×n
∗
∗
∗
QH
2 A Q2 = A3 + iA4 ,
∗
∗
∗
QH
1 B Q1 = B1 + iB2 ,
Ai , Bi ∈ Rn×n , i = 1, 2, 3, 4.
K)*]$\Æ!fk=G9, `, 9*:U&:
=Q
A
M1
0
0
M2
H
Q ,
=Q
B
(2.9)
N1
0
0
N2
QH ,
(2.10)
A& M , N , i =⎛1, 2 + (2.1),⎞ (2.2) J;1, `:
237
3
i
i
⎞
⎛
vec(M11 )
vec(A∗1 )
⎟
⎟
⎜
⎜
∗ ⎟
⎜vec(M12 )⎟
⎜
⎟ = W1 (I − G+ G1 )(W1 (I − G+ G1 ))+ ⎜vec(A2 ) ⎟ ,
⎜
1
1
⎜ vec(N ) ⎟
⎜vec(B ∗ )⎟
11 ⎠
⎝
⎝
1 ⎠
vec(N12 )
vec(B2∗ )
⎛
⎞
⎞
⎛
vec(M21 )
vec(A∗3 )
⎜
⎟
⎟
⎜
∗ ⎟
⎜vec(M22 )⎟
⎜
⎜
⎟ = W2 (I − G+ G2 )(W2 (I − G+ G2 ))+ ⎜vec(A4 ) ⎟ .
2
2
⎜ vec(N ) ⎟
⎜vec(B ∗ )⎟
21 ⎠
⎝
⎝
3 ⎠
vec(N22 )
vec(B4∗ )
*?
J*h )*&!Z[
K?fk=G;)*h )*]$\Æ!fk=G9
.
(2.6), (2.7), (2.8)
,
, 2P]V,
,
SA,B
.
(`a C
n×n
× C n×n
(2.11)
(2.12)
&!djZ,
fk=G9. *#;! A∗, B∗ ∈ C n×n, A, B ∈ SA,B , `? (2.9) J*h
2 2 2
f 2 (A∗ B ∗ ) − (A B) = A∗ − A + B ∗ − B 2 2
0
M1
N
∗
1
= A − Q
QH + B ∗ − Q
QH 0 N2
0 M2
2 2
0 M1
N1 0 = QH A∗ Q −
+ QH B ∗ Q −
0 M2 0 N2 2 QH A∗ Q QH A∗ Q
QH B ∗ Q QH B ∗ Q
M1
N1
0 1
2
1
2
1
1
1
1
=
−
−
+
∗
H ∗
H ∗
H ∗
QH
0
M
0
A
Q
Q
A
Q
Q
B
Q
Q
B
Q
1
2
2
1
2
2
2
2
2
H ∗
2 H ∗
2 H ∗ 2 H ∗ 2
= Q1 A Q1 − M1 + Q2 A Q2 − M2 + Q1 A Q2 + Q2 A Q1 2 H ∗
∗
+ Q B Q2 − N2 2 + QH B ∗ Q2 2 + QH B ∗ Q1 2
+ QH
1 B Q1 − N1
2
1
2
2 2 2
∗
= (A∗1 − M11 ) + i(A∗2 − M12 ) + (A∗3 − M21 ) + i(A∗4 − M22 ) + QH
1 A Q2
H ∗ 2 ∗
2
2
+ Q2 A Q1 + (B1 − N11 ) + i(B2∗ − N12 ) + (B3∗ − N21 ) + i(B4∗ − N22 )
2 H ∗ 2
∗
+ Q2 B Q1 .
+ QH
1 B Q2
? F W#!85-f F W#> 2 W#!.!*45
2
N2 2 2 2 2 2
∗
f 2 = A∗1 − M11 + A∗2 − M12 + A∗3 − M21 + A∗4 − M22 + QH
1 A Q2
H ∗ 2 ∗
2 ∗
2 ∗
2 ∗
2
+ Q2 A Q1 + B1 − N11 + B2 − N12 + B3 − N21 + B4 − N22 2 H ∗ 2
∗
+ Q B Q1 + QH
1 B Q2
2
2 2 2 2
∗
= vec(A1 − M11 ) 2 + vec(A∗2 − M12 )2 + vec(A∗3 − M21 )2 + vec(A∗4 − M22 )2
2 2 2 2
+ vec(B1∗ − N11 )2 + vec(B2∗ − N12 )2 + vec(B3∗ − N21 )2 + vec(B4∗ − N22 )2
2 H ∗ 2 H ∗ 2 H ∗ 2
∗
+ Q2 A Q1 + Q1 B Q2 + Q2 B Q1 + QH
1 A Q2
⎛
⎛
⎞
⎞
vec(A∗ − M ) 2 vec(A∗ − M ) 2
11
21
1
3
⎟
⎟
⎜
⎜
H ∗ 2 H ∗ 2
⎜vec(A∗2 − M12 )⎟
⎜vec(A∗4 − M22 )⎟
⎟
⎟
⎜
⎜
=
⎜ vec(B ∗ − N ) ⎟ + ⎜ vec(B ∗ − N ) ⎟ + Q1 A Q2 + Q2 A Q1
11 ⎠
21 ⎠
⎝
⎝
1
3
vec(B2∗ − N12 ) vec(B4∗ − N22 ) 2
2
2 H ∗ 2
∗
+ QH
B
Q
+
B
Q
Q
2
1
1
2
0
238
WONWOXO
2010
⎛
⎞ ⎛
⎞2 ⎛
⎞ ⎛
⎞2
vec(A∗ )
vec(A∗ )
vec(M
vec(M
)
)
11
21
1
3
⎜
⎜
⎜
⎜
⎟
⎟
⎟
⎟
⎜vec(A∗2 ) ⎟ ⎜vec(M12 )⎟
⎜vec(A∗4 ) ⎟ ⎜vec(M22 )⎟
⎜
⎜
⎜
⎜
⎟
⎟
⎟
⎟
−⎜
= ⎜
⎟ + ⎜vec(B ∗ )⎟ − ⎜ vec(N ) ⎟
∗ ⎟
⎝vec(B1 )⎠ ⎝ vec(N11 ) ⎠
⎝
21 ⎠
⎝
3 ⎠
vec(B2∗ )
vec(B4∗ )
vec(N12 ) vec(N22 ) 2
2 H ∗ 2 2 H ∗ 2 H ∗ 2
∗
+ Q2 A Q1 + Q1 B Q2 + Q2 B Q1 + QH
1 A Q2
⎛
2 ⎛
2
⎞
⎞
vec(A∗ )
vec(A∗ )
1
3
⎜
⎜
⎟
⎟
∗
∗
⎜
⎜vec(A2 ) ⎟
⎟
vec(A
)
4 ⎟
+
+
⎜
⎜
⎟
− W1 (I − G1 G1 )x + ⎜
− W2 (I − G2 G2 )y = ⎜
∗ ⎟
∗ ⎟
⎝vec(B1 )⎠
⎝vec(B3 )⎠
vec(B2∗ )
vec(B4∗ )
2
2
H ∗ 2 H ∗ 2 H ∗ 2 H ∗ 2
+ Q1 A Q2 + Q2 A Q1 + Q1 B Q2 + Q2 B Q1
9(, ? (2.13) J*h, i)* (A,B)∈S
min
∗
A,B
(A
∗
B ) − (A
B)
,:9,2c
⎛
⎞
vec(A∗ )
1
⎜
⎟
⎜vec(A∗2 ) ⎟
+
⎟
⎜
min ⎜
− W1 (I − G1 G1 )x
,
x vec(B ∗ )⎟
⎝
1 ⎠
vec(B2∗ )
2
⎛
⎞
vec(A∗ )
3
⎜
⎟
⎜vec(A∗4 ) ⎟
+
⎜
⎟
− W2 (I − G2 G2 )y min ⎜
.
y vec(B ∗ )⎟
⎝
3 ⎠
vec(B4∗ )
K?D) 3(2) *h, (2.14), (2.15) J!9*1B:U&
)*
(2.13)
(2.14)
(2.15)
2
+
+
x
= x0 + (I − (W1 (I − G+
1 G1 )) (W1 (I − G1 G1 )))x1 ,
⎛
⎞
vec(A∗1 )
⎜
⎟
∗ ⎟
⎜
+
+ ⎜ vec(A2 )⎟
.
x0 = (W1 (I − G1 G1 )) ⎜
∗ ⎟
⎝vec(B1 )⎠
vec(B2∗ )
+
+
y = y0 + (I − (W2 (I − G+
2 G2 )) (W2 (I − G2 G2 )))y1 ,
⎛
⎞
vec(A∗3 )
⎜
⎟
∗ ⎟
⎜
+
+ ⎜ vec(A4 ) ⎟
.
y0 = (W2 (I − G2 G2 )) ⎜
∗ ⎟
⎝vec(B3 )⎠
vec(B4∗ )
(2.16)
(2.17)
,C x1 ∈ C 2r(r+1), y1 ∈ C 2(n−r)(n−r+1) (e g! Y 5. 1BS (2.16), (2.17) J e R (2.7),
(2.8) J D*45 (1.4) J !9. Xf.
(+,
jk l m
?;) 2 *h, *4Æ9fk=G)*9!^3+,: #; P, A∗ .B∗, X, Λ.
(1) Y (2.9) JÆ $ A∗1 , A∗2 , A∗3 , A∗4 , B1∗ , B2∗ , B3∗ , B4∗ ,
3.
3
239
Y (1.1),(1.2) JÆ$ Kr , Kn−r , Tr , Tn−r ,
(3) ? X, Y (2.3) JÆ $ X1 , X2 ,
(4) Y (2.4) J ' ^ $ R1 , R2 , R3 , R4 , G1 , W1 , Y (2.5) J ' ^ $ R5 , R6 , R7 , R8 , G2 , W2 ,
(5) Y (2.11) JÆ 4 (vec(M11 ) vec(M12 ) vec(N11 ) vec(N12 ))T , Y (2.12) JÆ 4
(vec(M21 ) vec(M22 ) vec(N21 ) vec(N22 ))T c
(6) 1BS Æ 4! Y 5li, . mY (2.1), (2.2) J 45-. M1 , M2 , N1 , N2 , fmne R
B.
(2.10) JÆ 4 )* & (1.4) J !fk=G0 A,
o. ,CE n = 4, r = 2, m = 4, `#;
(2)
⎞
−0.8944 − 0.4472i
0
⎟
⎜
⎜
0
−0.8944 − 0.4472i⎟
⎟,
⎜
P =⎜
⎟
0
0
0
⎠
⎝−0.8944 + 0.4472i
0
−0.8944 + 0.4472i
0
0
⎛
0
0
0
0
⎞
1.0000 0 + 2.0000i
1.0000
2.0000
⎟
⎜
⎜ 0
1.0000
1.0000
0 + 2.0000i⎟
∗
⎟,
⎜
A =⎜
0 + 1.0000i
2.0000
0 + 1.0000i⎟
⎠
⎝ 0
2.0000
1.0000
0 + 1.0000i
1.0000
⎞
⎛
1.0000
0 + 1.0000i
2.0000
2.0000
⎟
⎜
⎜ 1.0000
0
0 + 3.0000i 2.0000⎟
∗
⎟,
⎜
B =⎜
2.0000
1.0000
0 ⎟
⎠
⎝0 + 1.0000i
1.0000
1.0000
0 + 3.0000i 2.0000
⎛
⎞
1.0000
2.0000 + 2.0000i
0 + 2.0000i
3.0000 + 1.0000i
⎜
⎟
⎜3.0000 + 2.0000i 2.0000 + 2.0000i 2.0000 + 1.0000i 1.0000 + 2.0000i⎟
⎜
⎟,
X=⎜
⎟
0 + 1.0000i
0
1.0000
⎝ 0 + 2.0000i
⎠
3.0000 + 2.0000i
1.0000
2.0000
2.0000
⎛
⎞
1 0 0 0
⎜
⎟
⎜0 1 0 0⎟
⎜
⎟.
(1.1), (1.2)
Kr , Kn−r , Tr , Tn−r
Λ=⎜
⎟
⎝0 0 2 0⎠
0 0 0 2
⎛
JÆ$!
Y
⎛
1 0
⎜
⎜0 1
K2 = ⎜
⎜0 1
⎝
0 0
Y%4gp45lim!9
&
0.1056 1.1736
M11 =
N12 =
1.1736 1.0528
0
0.2854
,
−0.2854
0
M12 =
,
M21 =
⎞
0
⎟
0⎟
⎟,
0⎟
⎠
1
1B&
⎛
0
0
0
⎞
⎜
⎟
⎜0 1 0 ⎟
⎜
⎟.
T2 = ⎜
⎟
⎝0 −1 0⎠
0 0 0
0
−0.0972
1.9472
−0.7264
0.0972
0
,
−0.7264
1.8944
N11 =
,
M22
−0.3416 1.2028
,
1.2028 0.3239
0
−0.7972
=
,
0.7972
0
WONWOXO
240
1.6708 −2.0972
,
−2.0972 2.3416
N21 =
N22 =
hD45
2010
0
0.3854
.
−0.3854
0
0.1056 − 0.0000i 1.1736 + 0.0972i
1.9472 + 0.0000i
−0.7264 − 0.7972
M1 =
, M2 =
,
1.1736 − 0.0972i 1.0528 − 0.0000i
−0.7264 + 0.7972i 1.8944 − 0.0000i
−0.3416 − 0.0000i 1.2028 − 0.2854i
1.6708 + 0.0000i −2.0972 + 0.3854i
N1 =
, N2 =
,
1.2028 + 0.2854i 0.3292 − 0.0000i
−2.0972 − 0.3854i 2.3416 + 0.0000i
fq45
⎛ )* !fk=G9 & i
0.5000 + 0.0000
⎜
⎜0.0000 − 0.5000i
=⎜
A
⎜0.4000 − 0.2000i
⎝
0.9500 − 0.3500i
⎛
1.0000 + 0.0000i
⎜
⎜
= ⎜0.2500 + 0.5000i
B
⎜0.6000 − 0.3000i
⎝
1.6500 + 0.0500i
[1]
⎞
0.9500 + 0.3500i
⎟
0.8000 + 0.4000i ⎟
⎟,
1.5000 + 0.0000i −0.0000 + 0.5000i⎟
⎠
0.0000 − 0.5000i
1.0000 − 0.0000i
⎞
0.6000 + 0.3000i 1.6500 − 0.0500i
⎟
0.9500 + 1.3500i 1.2000 + 0.6000i⎟
⎟.
1.000 + 0.0000i 0.2500 − 0.5000i⎟
⎠
0.2500 + 0.5000i 1.0000 − 0.0000i
−0.0000 + 0.5000i 0.4000 + +0.2000i
1.0000 − 0.0000i
0.8500 + 0.5500i
0.8500 − 0.5500i
0.8000 − 0.4000i
0.2500 − 0.5000i
1.0000 − 0.0000i
0.9500 − 1.3500i
1.2000 − 0.6000i
r s t u
jkl, kvn, olm. mp, Oqnrowxs. n" p
o
t, pl, 1986 8 .
[2] Baruch M. Optimization procedure to correct stiffness and flexibility matrices using vibration
q. q
rr
[J]. WO, 1990, 2: 177-187.
uss, vty, ow. qrxr [J]. WONWO XO,
tests[J]. AIAA J, 1978, 16: 1208-1210.
[3]
[4]
uzv. wytu [J]. WO, 1993, 3: 310-317.
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[5]
[6] Dai H. An algorithm for symmetric generalized inverse eigenvalue problems[J]. Linear Algebra
Appl., 1999, 296: 79-98.
[7] Dai H, Lancaster P. Newton’s method for a generalized inverse eigenvalue problems[J]. Number,
{x, yvv. rxr
[J]. z{w (x), 2006, 44(3): 185-
Linear Algebra Appl., 1997, 4: 1-21.
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188.
[9] Zhang Z Z, Hu X Y and Zhang L. The solvability conditions for the inverse eigenvalue problem
|zy, |l{, }}. tz AXB + CY D = E r~|{||} [J]. WO, 2007,
of Herimitian-generallized Hamiltonian matrices[J]. Inverse Problems, 2002, 18: 1369-1376.
[10]
29(2): 203-215.
[11] Lijun Zhao, Xiyan Hu, Lei Zhang. Linear restriction problem of Hermitian reflexive matrices and
{}~, q. ~
[M]. : 
~x, 1991.
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[12]