Research Article An Inverse Eigenvalue Problem for Jacobi Matrices

Mathematical Problems in Engineering Volume 2011 Article ID 571781 11 pages doi:10.1155/2011/571781 Research Article An Inverse Eigenvalue Problem for Jacobi Matrices Zhengsheng Wang 1 and Baoiang Zhong 2 1 Department of Mathematics Naning University of Aeronautics and Astronautics Naning 210016 China 2 School of Computer Science and Technology Soochow University Suzhou 215006 China Correspondence should be addressed to Zhengsheng Wang wangzhengsheng@nuaa.edu.cn Received 25 November 2010; Revised 26 February 2011; Accepted 1 April 2011 Academic Editor: Jaromir Horace Copyright q 2011 Z. Wang and B. Zhong. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original wor is properly cited. A ind of inverse eigenvalue problem is proposed which is the reconstruction of a Jacobi matrix by given four or five eigenvalues and corresponding eigenvectors. The solvability of the problem is discussed and some sufficient conditions for existence of the solution of this problem are proposed. 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: a 1 b 1 b 1 a 2 b 2 b 2 a 3 b 3 J......... b i > 0. 1.1 b n 2 a n 1 b n 1 b n 1 a n A Jacobi matrix inverse eigenvalue problem roughly speaing is how to determine the elements of Jacobi matrix from given eigen data. This ind of problem has great value for many applications including vibration theory and structural design for example the vibrating 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 matrix from 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 andη supposed λ > μ > ξ > η and four real orthogonal vectors of size nx x 2... T y y 1 y 2...y n T m m 1 m 2...m n T r r 1 r 2...r n 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 x 2... T y y 1 y 2...y n T z z 1 z 2...z n T mm 1 m 2...m n T rr 1 r 2...r n T finding a Jacobi matrix J of size n such that λ x μ y ν z ξ m and η r areitsfiveeigenpairs. In Sections 2 and 3 thesufficient conditions for the existence and uniqueness of the solution of Problems 1 and 2 are derived respectively. Numerical algorithms and two numerical examples are given in Section 4. We give conclusion and remars 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 real orthognal vectors of size n x x 2... T yy 1 y 2...y n T there is a unique Jacobi matrix J such that λ x μ y are its two eigenpairs if the following condition is satisfied: d D > 0 1 2...n 1 2.1 where D d x i y i 1 2...n x y i1 y 1 / 0 1 2...n 1. 2.2 And the elements of matrix J are b ( λ μ ) d D 1 2...n 1 a 1 λ b 1x 2

Mathematical Problems in Engineering 3 a n λ b n 1 1 λ b 1x 1 b x / 0 x a ( ) b 1 y 1 b y 1 μ x 0 y 2 3...n 1. 2.3 From Lemma 2.1 we can see that under some conditions two eigenpairs can determine a unique Jacobi matrix. Therefore for Problem 1 we only prove that the Jacobi matrices determined 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 solution of Problem 1. Theorem 2.2. Problem 1 has a unique solution if the following conditions are satisfied: i λ μd 1 /D1 λ ξd 2 /D2 λ ηd 3 /D3 > 0; ii if x 0 thenλ μd 1 where /D 1 μ ξd 4 /D 4 μ ηd 5 /D 5 1 d 1 d 4 x i y i i1 y i m i i1 d 2 d 5 x i m i i1 y i r i i1 d 3 d 6 x i r i i1 m i r i i1 1 2...n 2.4 D 1 y x y 1 D2 m x m 1 D3 r x r 1 D 4 m y m 1 y 1 D5 r y r 1 y 1 D6 r m r 1 m 1 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: b ( ) 1 λ μ d 1 2...n 1 D 1 a 1 λ b 1x 2 a n λ b n 1 1 λ b 1x 1 b x / 0 x a ( ) b 1 y 1 b y 1 μ x 0 y 2 3...n 1 2.6 b λ ξd2 1 2...n 1 D 2 a 1 λ b 1x 2 a n λ b n 1 1 λ b 1x 1 b x / 0 a x 2 3...n 1 ξ b 1m 1 b m 1 x 0 m ( ) 3 λ η d b 1 2...n 1 D 3 2.7 a 1 λ b 1x 2 a n λ b n 1 1 2.8 λ b 1x 1 b x / 0 a x η b 1r 1 b r 1 x 0 r 2 3...n 1. From the conditions we have b b b > 0 1 2...n 1. 2.9

Mathematical Problems in Engineering 5 If x / 0 we have a a a ;ifx 0 ( λ μ 1 D 1 ( ) 1 λ μ d 1 D 1 1 ( λ μ 1 D 1 ( ) 1 λ μ d 1 D 1 1 ( μ ξ 4 D 4 ( ) 4 μ ξ d 1 D 4 1 ( μ η 4 D 4 ( ) 4 μ η d 1. D 4 1 2.10 Since 2.6 wehave b D 4 ( μ ξ 4 b 1 D 4 1 ( μ ξ 4 1. 2.11 That is ( μ ξ ) y m b 1 D 4 1 b D 4 0. 2.12 Since D i / 0andx 0 we have y / 0m / 0. D 4 1 m 1y m y 1 D 4 m y 1 m 1 y replacing D 4 1 D4 in 2.12thenwe have μ ( b 1 y 1 b y 1 ) y ξ b 1m 1 b m 1 m. 2.13 Thus if x 0 we also have a a. In the same way we have a a. Then a a a. Therefore J J J 2.14 with four eigenpairs λ x μ y ξ m andη 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)andthree real orthogonal vectors of size nx x 2... T y y 1 y 2...y n T z z 1 z 2...z n T there is a unique Jacobi matrix J such that λ x μ y ν z are its three eigenpairs if the following conditions are satisfied: i λ μd 1 /D1 ii if x 0 λ μd 1 /D 1 λ νd 2 /D2 > 0; μ νd 3 /D 3 1where d 1 x i y i i1 d 2 x i z i i1 d 3 y i z i i1 D 1 y x y 1 D2 z x z 1 D3 z y z 1 y 1 1 2...n 1. 3.1 And the elements of matrix J are b ( λ μ ) d D 1 2...n 1 a 1 λ b 1x 2 a n λ b n 1 1 3.2 λ b 1x 1 b x / 0 x a ( ) b 1 y 1 b y 1 μ x 0 y 2 3...n 1. From Lemma 3.1 we can see that under some conditions three eigenpairs can determine a unique Jacobi matrix. Therefore for Problem 2 we only prove that the Jacobi matrices determined 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 solution of Problem 2.

Mathematical Problems in Engineering 7 Theorem 3.2. Problem 2 has a unique solution if the following conditions are satisfied: ηd 7 i λ μd 1 /D1 ii if x /D 7 1 where λ νd 2 /D2 0 then λ μd 1 /D 1 λ ξd 3 /D3 μ νd 5 /D 5 λ ηd 4 /D4 μ ξd 6 > 0; μ /D 6 D 1 D 4 D 7 d 1 d 4 d 7 y x n y n y x i y i i1 x i n i i1 y i n i i1 y 1 n 1 y 1 n 1 y 1 D 10 d 10 d 2 d 5 d 8 m i n i n m i1 D2 D5 D8 n 1 m 1 x i z i i1 y i z i i1 z i m i i1 z x z y m z d 3 d 6 d 9 1 2...n z 1 z 1 y 1 m 1 z 1 D3 D6 D9 x i m i i1 y i m i i1 z i n i i1 1 2...n 1. m x m y n z m 1 m 1 y 1 n 1 z 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: b ( ) 1 λ μ d 1 2...n 1 D 1 a 1 λ b 1x 2 a n λ b n 1 1 λ 1x 1 b x x / 0 a ( ) b 1 y 1 b y 1 μ y x 0 2 3...n 1

8 Mathematical Problems in Engineering ( ) 1 λ μ d b 1 2...n 1 D 1 a 1 λ b 1x 2 a n λ b n 1 1 λ b 1x 1 b x / 0 x a ( ) b 1 y 1 b y 1 μ x 0 y 2 3...n 1 ( ) 1 λ μ d b 1 2...n 1 D 1 a 1 λ b 1x 2 a n λ b n 1 1 λ b 1x 1 b x / 0 x a ( ) b 1 y 1 b y 1 μ x 0 y 2 3...n 1 3.4 From conditions i and ii we have obviously b b b > 0 1 2...n 1 a a a. 3.5 Therefore J J J 3.6 with five eigenpairs λ x μ y ν z ξ m andη r. 4. Numerical Algorithms and Examples The process of the proof of the theorem provides us with a recipe for finding the solution of Problem 1 if it exists. From Theorem 2.2 we propose a numerical algorithm for finding the unique solution of Problem 1 as follows.

Mathematical Problems in Engineering 9 Algorithm 1. Input. The real numbers λ > μ > ξ > η and mutually orthogonal vectors x y m r. Output. The symmetric Jacobi matrix having the eigenpairs λ x μ y ξ m η r: 1 compute d 1 d2 2 if any one of D 1 d3 D2 by this method; 3 for 1 2...n 1. a When x 0 if d4 D3 d5 d6 D4 and D 1 D2 D3 D4 D5 D6 ; D5 D6 is zero the Problem 1 can not be solved ( λ μ 1 D 1 ( μ ξ 4 D 4 ( μ η 5 D 5 1 4.1 then a μ ( ) 1 λ μ d b D 1 ( ) b 1 y 1 b y 1 y. 4.2 Otherwise Problem 1 has no solution. b When x / 0 if ( λ μ 1 D 1 λ ξd2 D 2 ( λ η 3 D 3 > 0 4.3 then b ( ) 1 λ μ d D 1 a λ b 1x 1 b x. 4.4 Otherwise Problem 1 has no solution; 4 a n λ b n 1 1 /. Note that we can also propose a numerical algorithm from Theorem 3.2. Becauseof the limitation of space we don t describe it here in detail. Now we give two numerical examples here to illustrate that the results obtained in this 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 vectors x 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 itis easy to verify that these given data satisfy the conditions of the Theorem 2.2. After calculating on the microcomputer through maing program of Algorithm 1 we have a unique Jacobi matrix: 2 1 1 2 1 J 1 2 1. 4.5 1 2 1 1 2 Example 4.2. Given five real numbers λ 7.543 μ 3.543 ν 2 ξ 4.296 and η 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 andr 0.4619 0.3536 0.1913 0.5000 0.1913 0.3536 0.4619 T it is easy to verify that these given numbers can not satisfy the conditions of the Theorem 2.2 but Theorem 3.2. After calculating on the microcomputer through maing program of Theorem 3.2 wehaveajacobimatrix: 2 3 3 2 3 3 2 3 J 3 2 3. 4.6 3 2 3 3 2 3 3 2 5. Conclusion and Remars As a summary we have presented some sufficient conditions as well as simple methods to construct a Jacobi matrix from its four or five eigenpairs. Numerical examples have been given to illustrate the effectiveness of our results and the proposed method. Also the idea in this paper may provide some insights for other banded matrix inverse eigenvalue problems. Acnowledgments This wor is supported by the NUAA Research funding Grant NS2010202 and the Aviation Science Foundation of China Grant 2009ZH52069. The authors would lie to than Professor 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 Inverse Problems vol. 13 no. 4 pp. 1071 1081 1997. 2 N. Radwan An inverse eigenvalue problem for symmetric and normal matrices Linear Algebra and Its Applications vol. 248 pp. 101 109 1996. 3 Z. S. Wang Inverse eigenvalue problem for real symmetric five-diagonal matrix Numerical Mathematics 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 Kluwer Academic 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 inverse eigenproblems of tridiagonal symmetric matrices Journal on Numerical Methods and Computer Applications vol. 21 no. 2 pp. 102 111 2000. 7 X. Y. Hu and X. Z. Zhou Inverse eigenvalue problems for symmetric tridiagonal matrices Journal on Numerical Methods and Computer Applications vol. 17 pp. 150 156 1996.

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