-
Hindawi Publishing CorporationMathematical Problems in
EngineeringVolume 2011, Article ID 571781, 11
pagesdoi:10.1155/2011/571781
Research ArticleAn Inverse Eigenvalue Problem for Jacobi
Matrices
Zhengsheng Wang1 and Baojiang Zhong2
1 Department of Mathematics, Nanjing University of Aeronautics
and Astronautics,Nanjing 210016, China
2 School of Computer Science and Technology, Soochow University,
Suzhou 215006, China
Correspondence should be addressed to Zhengsheng Wang,
[email protected]
Received 25 November 2010; Revised 26 February 2011; Accepted 1
April 2011
Academic Editor: Jaromir Horacek
Copyright q 2011 Z. Wang and B. Zhong. This is an open access
article distributed under theCreative Commons Attribution License,
which permits unrestricted use, distribution, andreproduction in
any medium, provided the original work is properly cited.
A kind of inverse eigenvalue problem is proposed which is the
reconstruction of a Jacobi matrixby given four or five eigenvalues
and corresponding eigenvectors. The solvability of the problemis
discussed, and some sufficient conditions for existence of the
solution of this problem areproposed. Furthermore, a numerical
algorithm and two examples are presented.
1. Introduction
An n × n matrix J is called a Jacobi matrix if it is of the
following form:
J �
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
a1 b1
b1 a2 b2
b2 a3 b3
. . . . . . . . .
bn−2 an−1 bn−1
bn−1 an
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
, bi > 0. �1.1�
A Jacobi matrix inverse eigenvalue problem, roughly speaking, is
how to determinethe elements of Jacobi matrix from given eigen
data. This kind of problem has great valuefor many applications,
including vibration theory and structural design, for example,
thevibrating rod model �1, 2�. In recent years, some new results
have been obtained on the
-
2 Mathematical Problems in Engineering
construction of a Jacobi matrix �3, 4�. However, the problem of
constructing a Jacobi matrixfrom its four or five eigenpairs has
not been considered yet. The problem is as follows.
Problem 1. Given four different real scalars λ, μ, ξ, and η
�supposed λ > μ > ξ > η�and four real orthogonal vectors
of size nx � �x1, x2, . . . , xn�
T , y � �y1, y2, . . . , yn�T , m �
�m1, m2, . . . , mn�T , r � �r1, r2, . . . , rn�T , finding a
Jacobi matrix J of size n such that�λ, x�, �μ, y�, �ξ,m�, and �η,
r� are its four eigenpairs.
Problem 2. Given five different real scalars λ, μ, ν, ξ, and η
�supposed λ > μ > ν > ξ >η� and five real orthogonal
vectors of size nx � �x1, x2, . . . , xn�T , y � �y1, y2, . . . ,
yn�T , z ��z1, z2, . . . , zn�T , m � �m1, m2, . . . , mn�T , r �
�r1, r2, . . . , rn�T , finding a Jacobi matrix J of size nsuch
that �λ, x�, �μ, y�, �ν, z�, �ξ, m�, and �η, r� are its five
eigenpairs.
In Sections 2 and 3, the sufficient conditions for the existence
and uniqueness ofthe solution of Problems 1 and 2 are derived,
respectively. Numerical algorithms and twonumerical examples are
given in Section 4. We give conclusion and remarks in Section
5.
2. The Solvability Conditions of Problem 1
Lemma 2.1 �see �5, 6��. Given two different real scalars λ, μ
�supposed λ > μ� and two realorthognal vectors of size n, x �
�x1, x2, . . . , xn�T , y � �y1, y2, . . . , yn�T , there is a
unique Jacobimatrix J such that �λ, x�, �μ, y� are its two
eigenpairs if the following condition is satisfied:
dkDk
> 0, �k � 1, 2, . . . , n − 1�, �2.1�
where
dk �k∑i�1
xiyi, �k � 1, 2, . . . , n�,
Dk �
∣∣∣∣∣xk xk1
yk yk1
∣∣∣∣∣/� 0, �k � 1, 2, . . . , n − 1�.
�2.2�
And the elements of matrix J are
bk �
(λ − μ)dkDk
, �k � 1, 2, . . . , n − 1�,
a1 � λ − b1x2x1
,
-
Mathematical Problems in Engineering 3
an � λ − bn−1xn−1xn
,
ak �
⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩
λ − �bk−1xk−1 bkxk1�xk
, xk /� 0,
μ −(bk−1yk−1 bkyk1
)
yk, xk � 0,
�k � 2, 3, . . . , n − 1�.
�2.3�
FromLemma 2.1, we can see that under some conditions two
eigenpairs can determinea unique Jacobi matrix. Therefore, for
Problem 1, we only prove that the Jacobi matricesdetermined by �λ,
x�, �μ, y� and �ξ,m�, �η, r� are the same.
The following theorem gives a sufficient condition for the
uniqueness of the solutionof Problem 1.
Theorem 2.2. Problem 1 has a unique solution if the following
conditions are satisfied:
�i� �λ − μ�d�1�k/D
�1�k
� �λ − ξ�d�2�k/D
�2�k
� �λ − η�d�3�k/D
�3�k
> 0;
�ii� if xk � 0, then �λ − μ�d�1�j /D�1�j � �μ − ξ�d
�4�j /D
�4�j � �μ − η�d
�5�j /D
�5�j , j � k, k − 1,
where
d�1�k
�k∑i�1
xiyi, d�2�k
�k∑i�1
ximi, d�3�k
�k∑i�1
xiri,
d�4�k
�k∑i�1
yimi, d�5�k
�k∑i�1
yiri, d�6�k
�k∑i�1
miri,
�k � 1, 2, . . . , n�, �2.4�
D�1�k �
∣∣∣∣∣yk yk1
xk xk1
∣∣∣∣∣, D�2�k �
∣∣∣∣∣mk mk1
xk xk1
∣∣∣∣∣, D�3�k �
∣∣∣∣∣rk rk1
xk xk1
∣∣∣∣∣,
D�4�k
�
∣∣∣∣∣mk mk1
yk yk1
∣∣∣∣∣, D�5�k
�
∣∣∣∣∣rk rk1
yk yk1
∣∣∣∣∣, D�6�k
�
∣∣∣∣∣rk rk1
mk mk1
∣∣∣∣∣,
�k � 1, 2, . . . , n − 1�.
�2.5�
Proof. According to Lemma 2.1, under certain condition, �λ, x�
and �μ, y�, �λ, x� and �ξ,m�,�λ, x� and �η, r� can determine one
unique Jacobi matrix, denoted J, J ′, J ′′, respectively. Their
-
4 Mathematical Problems in Engineering
elements are as follows:
bk �
(λ − μ)d�1�
k
D�1�k
, �k � 1, 2, . . . , n − 1�,
a1 � λ − b1x2x1
,
an � λ − bn−1xn−1xn
,
ak �
⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩
λ − �bk−1xk−1 bkxk1�xk
, xk /� 0,
μ −(bk−1yk−1 bkyk1
)
yk, xk � 0,
�k � 2, 3, . . . , n − 1�,
�2.6�
b′k ��λ − ξ�d�2�k
D�2�k
, �k � 1, 2, . . . , n − 1�,
a′1 � λ −b1x2x1
,
a′n � λ −bn−1xn−1
xn,
a′k �
⎧⎪⎪⎪⎨⎪⎪⎪⎩
λ − �bk−1xk−1 bkxk1�xk
, xk /� 0,
ξ − �bk−1mk−1 bkmk1�mk
, xk � 0,
�k � 2, 3, . . . , n − 1�,
�2.7�
b′′k �
(λ − η)d�3�
k
D�3�k
, �k � 1, 2, . . . , n − 1�,
a′′1 � λ −b1x2x1
,
a′′n � λ −bn−1xn−1
xn,
a′′k �
⎧⎪⎪⎪⎨⎪⎪⎪⎩
λ − �bk−1xk−1 bkxk1�xk
, xk /� 0,
η − �bk−1rk−1 bkrk1�rk
, xk � 0,
�k � 2, 3, . . . , n − 1�.
�2.8�
From the conditions, we have
bk � b′k � b′′k > 0, k � 1, 2, . . . , n − 1. �2.9�
-
Mathematical Problems in Engineering 5
If xk /� 0, we have ak � a′k � a′′k; if xk � 0,
(λ − μ)d�1�
k
D�1�k
�
(μ − ξ)d�4�
k
D�4�k
,
(λ − μ)d�1�
k−1D
�1�k−1
�
(μ − ξ)d�4�
k−1D
�4�k−1
,
(λ − μ)d�1�kD
�1�k
�
(μ − η)d�4�kD
�4�k
,
(λ − μ)d�1�
k−1D
�1�k−1
�
(μ − η)d�4�
k−1D
�4�k−1
.
�2.10�
Since �2.6�, we have
bkD�4�k
�(μ − ξ)d�4�
k,
bk−1D�4�k−1 �
(μ − ξ)d�4�
k−1.�2.11�
That is,
(μ − ξ)ykmk bk−1D�4�k−1 − bkD
�4�k � 0. �2.12�
Since D�i�k /� 0 and xk � 0, we have yk /� 0, mk /� 0.D
�4�k−1 � mk−1yk −mkyk−1, D
�4�k
� mkyk1 −mk1yk replacingD�4�k−1, D�4�k
in �2.12�, then wehave
μ −(bk−1yk−1 bkyk1
)
yk� ξ − �bk−1mk−1 bkmk1�
mk. �2.13�
Thus, if xk � 0, we also have ak � a′k. In the same way, we have
ak � a′′k. Then, ak � a′k � a
′′k.
Therefore,
J � J ′ � J ′′ �2.14�
with four eigenpairs �λ, x�, �μ, y�, �ξ,m�, and �η, r�.
-
6 Mathematical Problems in Engineering
3. The Solvability Conditions of Problem 2
Lemma 3.1 �see �7��. Given three different real scalars λ, μ, ν
(supposed λ > μ > ν) and threereal orthogonal vectors of size
nx � �x1, x2, . . . , xn�
T , y � �y1, y2, . . . , yn�T , z � �z1, z2, . . . , zn�
T ,there is a unique Jacobi matrix J such that �λ, x�, �μ, y�,
�ν, z� are its three eigenpairs if the followingconditions are
satisfied:
�i� �λ − μ�d�1�k/D
�1�k
� �λ − ν�d�2�k/D
�2�k
> 0;
�ii� if xk � 0, �λ − μ�d�1�j /D�1�j � �μ − ν�d
�3�j /D
�3�j , j � k, k − 1, where
d�1�k �
k∑i�1
xiyi, d�2�k �
k∑i�1
xizi, d�3�k �
k∑i�1
yizi,
D�1�k
�
∣∣∣∣∣yk yk1
xk xk1
∣∣∣∣∣, D�2�k
�
∣∣∣∣∣zk zk1
xk xk1
∣∣∣∣∣, D�3�k
�
∣∣∣∣∣zk zk1
yk yk1
∣∣∣∣∣,
�k � 1, 2, . . . , n − 1�. �3.1�
And the elements of matrix J are
bk �
(λ − μ)dkDk
�k � 1, 2, . . . , n − 1�,
a1 � λ − b1x2x1
,
an � λ − bn−1xn−1xn
,
ak �
⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩
λ − �bk−1xk−1 bkxk1�xk
, xk /� 0,
μ −(bk−1yk−1 bkyk1
)
yk, xk � 0,
�k � 2, 3, . . . , n − 1�.
�3.2�
From Lemma 3.1, we can see that under some conditions three
eigenpairs can determinea unique Jacobi matrix. Therefore, for
Problem 2, we only prove that the Jacobi matricesdetermined by �λ,
x�, �μ, y�, �ν, z�; �λ, x�, �μ, y�, �ξ,m�, �λ, x�, �μ, y�, �η, r�
are the same.
The following theorem gives a sufficient condition for the
uniqueness of the solutionof Problem 2.
-
Mathematical Problems in Engineering 7
Theorem 3.2. Problem 2 has a unique solution if the following
conditions are satisfied:
�i� �λ − μ�d�1�k /D�1�k � �λ − ν�d
�2�k /D
�2�k � �λ − ξ�d
�3�k /D
�3�k � �λ − η�d
�4�k /D
�4�k > 0;
�ii� if xk � 0, then �λ − μ�d�1�j /D�1�j � �μ − ν�d
�5�j /D
�5�j � �μ − ξ�d
�6�j /D
�6�j � �μ −
η�d�7�j /D�7�j , j � k, k − 1, where
d�1�k
�k∑i�1
xiyi, d�2�k
�k∑i�1
xizi, d�3�k
�k∑i�1
ximi,
d�4�k
�k∑i�1
xini, d�5�k
�k∑i�1
yizi, d�6�k
�k∑i�1
yimi,
d�7�k
�k∑i�1
yini, d�8�k
�k∑i�1
zimi, d�9�k
�k∑i�1
zini,
d�10�k
�k∑i�1mini, �k � 1, 2, . . . , n�
D�1�k
�
∣∣∣∣∣yk yk1
xk xk1
∣∣∣∣∣, D�2�k
�
∣∣∣∣∣zk zk1
xk xk1
∣∣∣∣∣, D�3�k
�
∣∣∣∣∣mk mk1
xk xk1
∣∣∣∣∣,
D�4�k
�
∣∣∣∣∣nk nk1
yk yk1
∣∣∣∣∣, D�5�k
�
∣∣∣∣∣zk zk1
yk yk1
∣∣∣∣∣, D�6�k
�
∣∣∣∣∣mk mk1
yk yk1
∣∣∣∣∣,
D�7�k �
∣∣∣∣∣nk nk1
yk yk1
∣∣∣∣∣, D�8�k �
∣∣∣∣∣mk mk1
zk zk1
∣∣∣∣∣, D�9�k �
∣∣∣∣∣nk nk1
zk zk1
∣∣∣∣∣,
D�10�k �
∣∣∣∣∣nk nk1
mk mk1
∣∣∣∣∣, �k � 1, 2, . . . , n − 1�.
�3.3�
Proof. According to Lemma 3.1, under certain condition, �λ, x�,
�μ, y�, �ν, z�; �λ, x�, �μ, y�,�ξ,m�, �λ, x�, �μ, y�, �η, r� can
determine one unique Jacobi matrix, denoted J, J ′, J
′′,respectively. Their elements are as follows:
bk �
(λ − μ)d�1�
k
D�1�k
�k � 1, 2, . . . , n − 1�,
a1 � λ − b1x2x1
,
an � λ − bn−1xn−1xn
,
ak �
⎧⎪⎪⎪⎨⎪⎪⎪⎩
λ − �bk−1xk−1 bkxk1�xk
, xk /� 0,
μ −(bk−1yk−1 bkyk1
)
yk, xk � 0,
�k � 2, 3, . . . , n − 1�,
-
8 Mathematical Problems in Engineering
b′k�
(λ − μ)d�1�
k
D�1�k
, �k � 1, 2, . . . , n − 1�,
a′1 � λ −b1x2x1
,
a′n � λ −bn−1xn−1
xn,
a′k�
⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩
λ − �bk−1xk−1 bkxk1�xk
, xk /� 0,
μ −(bk−1yk−1 bkyk1
)
yk, xk � 0,
�k � 2, 3, . . . , n − 1�,
b′′k�
(λ − μ)d�1�kD
�1�k
, �k � 1, 2, . . . , n − 1�,
a′′1 � λ −b1x2x1
,
a′′n � λ −bn−1xn−1
xn,
a′′k�
⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩
λ − �bk−1xk−1 bkxk1�xk
, xk /� 0,
μ −(bk−1yk−1 bkyk1
)
yk, xk � 0,
�k � 2, 3, . . . , n − 1�,
�3.4�
From conditions �i� and �ii� we have obviously
bk � b′k � b′′k > 0, k � 1, 2, . . . , n − 1, ak � a′k �
a′′k. �3.5�
Therefore,
J � J ′ � J ′′ �3.6�
with five eigenpairs �λ, x�, �μ, y�, �ν, z�, �ξ,m�, and �η,
r�.
4. Numerical Algorithms and Examples
The process of the proof of the theorem provides us with a
recipe for finding the solution ofProblem 1 if it exists.
From Theorem 2.2, we propose a numerical algorithm for finding
the unique solutionof Problem 1 as follows.
-
Mathematical Problems in Engineering 9
Algorithm 1. Input. The real numbers λ > μ > ξ > η and
mutually orthogonal vectorsx, y,m, r.
Output. The symmetric Jacobi matrix having the eigenpairs �λ,
x�, �μ, y�, �ξ,m�, �η, r�:
�1� compute d�1�k, d
�2�k, d
�3�k, d
�4�k, d
�5�k, d
�6�k
and D�1�k, D
�2�k, D
�3�k, D
�4�k, D
�5�k, D
�6�k;
�2� if any one of D�1�k, D
�2�k, D
�3�k, D
�4�k, D
�5�k, D
�6�k
is zero, the Problem 1 can not be solvedby this method;
�3� for k � 1, 2, . . . , n − 1.�a� When xk � 0, if
(λ − μ)d�1�jD
�1�j
�
(μ − ξ)d�4�jD
�4�j
�
(μ − η)d�5�jD
�5�j
, j � k, k − 1, �4.1�
then
bk �
(λ − μ)d�1�
k
D�1�k
,
ak � μ −(bk−1yk−1 bkyk1
)
yk.
�4.2�
Otherwise, Problem 1 has no solution.
�b� When xk /� 0, if
(λ − μ)d�1�kD
�1�k
��λ − ξ�d�2�k
D�2�k
�
(λ − η)d�3�kD
�3�k
> 0, �4.3�
then
bk �
(λ − μ)d�1�kD
�1�k
,
ak � λ − �bk−1xk−1 bkxk1�xk
.
�4.4�
Otherwise, Problem 1 has no solution;
�4� an � λ − bn−1xn−1/xn.
Note that we can also propose a numerical algorithm from Theorem
3.2. Because ofthe limitation of space, we don’t describe it here
in detail.
Now we give two numerical examples here to illustrate that the
results obtained inthis paper are correct.
-
10 Mathematical Problems in Engineering
Example 4.1. Given four real numbers λ � 3, μ � 2, ξ � 1, η �
0.2679, and the four vectorsx � �1, 1, 0,−1,−1�T , y � �1, 0,−1, 0,
1�T , m � �1,−1, 0, 1,−1�T , r � �1,−√3, 2,−√3, 1�T , it iseasy to
verify that these given data satisfy the conditions of the Theorem
2.2. After calculatingon the microcomputer through making program
of Algorithm 1, we have a unique Jacobimatrix:
J �
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣
2 1
1 2 1
1 2 1
1 2 1
1 2
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦. �4.5�
Example 4.2. Given five real numbers λ � 7.543, μ � −3.543, ν �
2, ξ � 4.296, andη � −0.296, and the five vectors: x � �0.1913,
0.3536, 0.4619, 0.5000, 0.4619, 0.3536, 0.1913�T ,y �
�0.1913,−0.3536, 0.4619,−0.5000, 0.4619,−0.3536, 0.1913�T , z �
�0.5000, 0,−0.5000, 0,0.5000, 0,−0.5000�T , m � �0.4619,
0.3536,−0.1913, 0.5000,−0.1913, 0.3536, 0.4619�T , and r ��0.4619,
−0.3536,−0.1913, 0.5000,−0.1913,−0.3536, 0.4619�T , it is easy to
verify that these givennumbers can not satisfy the conditions of
the Theorem 2.2 but Theorem 3.2. After calculatingon the
microcomputer through making program of Theorem 3.2, we have a
Jacobi matrix:
J �
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
2 3
3 2 3
3 2 3
3 2 3
3 2 3
3 2 3
3 2
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
. �4.6�
5. Conclusion and Remarks
As a summary, we have presented some sufficient conditions, as
well as simple methods toconstruct a Jacobi matrix from its four or
five eigenpairs. Numerical examples have beengiven to illustrate
the effectiveness of our results and the proposed method. Also, the
idea inthis paper may provide some insights for other banded matrix
inverse eigenvalue problems.
Acknowledgments
This work is supported by the NUAA Research funding �Grant
NS2010202� and theAviation Science Foundation of China �Grant
2009ZH52069�. The authors would like to thankProfessor Hua Dai for
his valuable discussions.
-
Mathematical Problems in Engineering 11
References
�1� P. Nylen and F. Uhlig, “Inverse eigenvalue problem:
existence of special spring-mass systems,” InverseProblems, vol.
13, no. 4, pp. 1071–1081, 1997.
�2� N. Radwan, “An inverse eigenvalue problem for symmetric and
normal matrices,” Linear Algebra andIts Applications, vol. 248, pp.
101–109, 1996.
�3� Z. S. Wang, “Inverse eigenvalue problem for real symmetric
five-diagonal matrix,” NumericalMathematics, vol. 24, no. 4, pp.
366–376, 2002.
�4� G. M. L. Gladwell, Inverse Problems in Vibration, vol. 119
of Solid Mechanics and Its Applications, KluwerAcademic Publishers,
Dordrecht, The Netherlands, 2nd edition, 2004.
�5� H. Dai, “Inverse eigenvalue problems for Jacobi matrices and
symmetric tridiagonal matrices,”Numerical Mathematics, vol. 12, no.
1, pp. 1–13, 1990.
�6� A. P. Liao, L. Zhang, and X. Y. Hu, “Conditions for the
existence of a unique solution for inverseeigenproblems of
tridiagonal symmetric matrices,” Journal on Numerical Methods and
ComputerApplications, vol. 21, no. 2, pp. 102–111, 2000.
�7� X. Y. Hu and X. Z. Zhou, “Inverse eigenvalue problems for
symmetric tridiagonal matrices,” Journal onNumerical Methods and
Computer Applications, vol. 17, pp. 150–156, 1996.
-
Submit your manuscripts athttp://www.hindawi.com
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
MathematicsJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttp://www.hindawi.com
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
Probability and StatisticsHindawi Publishing
Corporationhttp://www.hindawi.com Volume 2014
Journal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
OptimizationJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
CombinatoricsHindawi Publishing
Corporationhttp://www.hindawi.com Volume 2014
International Journal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing
Corporationhttp://www.hindawi.com Volume 2014
International Journal of Mathematics and Mathematical
Sciences
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
The Scientific World JournalHindawi Publishing Corporation
http://www.hindawi.com Volume 2014
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com
Volume 2014 Hindawi Publishing Corporationhttp://www.hindawi.com
Volume 2014
Stochastic AnalysisInternational Journal of