Inverse Eigenvalue Problem with Non-simple Eigenvalues for Damped Vibration Systems

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1 Journal of Informatics Mathematical Sciences Volume 1 (2009), Numbers 2 & 3, pp RGN Publications (Invited paper) Inverse Eigenvalue Problem with Non-simple Eigenvalues for Damped Vibration Systems M. Mohseni Moghadam A. Tajaddini Abstract. In this paper, we will present a general form of real symmetric n n matrices M, C K for a quadratic inverse eigenvalue problem QIEP: Q(λ) (λ 2 M + λc + K)x 0, so that Q(λ) has a prescribed set of k eigenvalues with algebraic multiplicity n i, i 1,, k (which 2n 1 + 2n n l + n l n k 2n). This paper generalizes the method of inverse problem for selfadjoint linear pencils, to self-adjoint quadratic pencils Q(λ). It is shown that this inverse problem involves certain free parameters. Via appropriate choice of free variables in the general form of QIEP, we solve a QIEP. 1. Introduction The problem of finding scalars λ nontrivial vector x n such that Q(λ)x (λ 2 M + λc + K)x 0, (1.1) where M, C K are given n n real matrices, is known as the quadratic eigenvalue problem QEP. The nonzero vectors x the corresponding scalars λ are called eigenvectors eigenvalues of the QEP, respectively. It is known that if the leading coefficient matrix M is nonsingular, then the quadratic pencil will have 2n eigenvalues over. A theoretical analysis of QEPs can be found in the book by written Gohberg, Lancaster Rodman [5]. Many applications, mathematical properties a variety of numerical techniques for the QEP are surveyed in the treatise by Tisseur Meerbergen [4]. Generally, in many mathematical modeling, there is a correspondence between the internal parameters the external behavior of a system from a priori known physical parameters such as mass, length, elasticity, inductance, capacitance, 2000 Mathematics Subject Classification.15-; 15-02; 15A29. Key words phrases.inverse problem; Quadratic form; Non-simple eigenvalues. This work has been partially supported by the Center of Excellence of Linear Algebra Optimization, Shahid Bahonar University of Kerman.

2 92 M. Mohseni Moghadam A. Tajaddini so on, is referred to a direct problem. In contrast, the inverse problem is to determine or estimate the parameters of the system according to its measured or expected behavior. The concern in the direct problem is to express the behavior in term of parameters whereas in the inverse problem concern is to express the parameters in terms of behavior. The inverse problem is as important as the direct problem in applications. The purpose of this paper is to make use of a parametrization method of inverse eigenvalue problems for self-adjoint linear pencils, that Chu, Datta, Lin u developed earlier [1], to self-adjoint quadratic pencils Q(λ). Recently, Kuo, Lin u [2], Chu u [3] proposed to parametrize a self-adjoint inverse eigenvalue problem. We shall limit a quadratic inverse eigenvalue problem QIEP to the following realm. Determine a general solution of real symmetric coefficients matrices M, C K so that the resulting QEP has a prescribed set of an eigenpair.more precisely, by a QIEP we refer to the following inverse problem: QIEP: Let (Λ, ) R 2n 2n R n 2n be a given pair of matrices, where Λ diag{λ [2] 1,..., λ[2], λ l l+1 I nl+1,..., λ k I nk }, (1.2) λ [2] αi I i ni β i I ni, i 1,..., l, (1.3) β i I ni α i I ni [ 1R, 1I,..., lr, li, l+1,..., k ], (1.4) where n i, i 1,, k are algebraic multiplicity corresponding to λ i ir, ii R n n i, i 1,..., l, i R n n i, i l+1,..., k, so is nonsingular. The true eigenvalues eigenvectors are readily identifiable by the transformation 1 In1 I R : diag 2 n1 1 Inl I,, nl, I ii n1 ii n1 2 ii nl ii nl+1,, I nk, (1.5) nl with i 1. That is, by defining Λ R H ΛR diag{λ 1 I n1, λ 2 I n1,, λ 2l 1 I nl, λ 2l I nl, λ 2l+1 I nl+1,, λ k I nk } 2n 2n, (1.6) R [ 1, 2,, 2l 1, 2l, 2l+1,, k ] n 2n. (1.7) The true (complex-valued) eigenvalues eigenvectors of the desired quadratic pencil Q(λ) can be induced from the pair (Λ, ) of real matrices. In this case, note that 2 j 1 jr + i ji, 2j jr i ji, λ 2j 1 α j + iβ j, λ 2 j α j iβ j, for j 1,, l, where as j λ j are all real-valued for j 2l + 1,, k. Find a general form for symmetric matrices M,C K which satisfy in the following eqnarray M 2 + C + K 0. (1.8)

3 Inverse Eigenvalue Problem with Non-simple Eigenvalues for Damped Vibration Systems 93 Now, we define a stard eigenpair for the quadratic pencil that introduced in [5]. Definition 1.1. A pair of matrices (Λ, ) R 2n 2n R n 2n is called a stard eigenpair for the quadratic pencil Q(λ) if only if the matrix Y : (1.9) is nonsingular the eqnarray holds. M 2 + C + K 0 2. Solving IQEP In this section, we shall solve the IQEP for a given matrix pair (Λ, ) R 2n 2n R n 2n as in (1.2) (1.4) under assumption is nonsingular. It is easy to see that Q(λ)x 0 if only if x L(λ) 0, (2.1) λx where C M K 0 L(λ) : λ. (2.2) M 0 0 M We want to construct a general form for M, C K. We first tell the theorem that Chu, Datta, Lin u [1] prove for self-adjoint linear pencils. Theorem 2.1. A self-adjoint linear pencil can have arbitrary eigenstructure with distinct eigenvalues linearly independent eigenvectors. Indeed, given an eigenstructure (, Λ), the solutions (A, B) form a subspace of dimensionality n in the product space R n n R n n can be parametrized by the diagonal matrix Γ via the following relationships A T Γ 1, B T ΓΛ 1. We consider the problem quadratic that its eigenvalues are non-simple so Y is nonsingular. In order to, we construct a general form for matrices M, C K, we make use of Lancaster linearization Chu, Datta, Lin u s proof idea. We know that Λ, Y is an eigenpair for linear pencil L(λ). Let us consider the linear pencil λa B where C M A (2.3) M 0

4 94 M. Mohseni Moghadam A. Tajaddini B K 0. (2.4) 0 M For the linear pencil λa B to have eigenstructure (Λ, Y ), it is necessary that I2n Λ2n T Y T B T Y T A T 0. (2.5) On the other h, it is trivial that I2n Λ2n T Λ T S 0, (2.6) S for any S R 2n 2n. Thus we obtain a parametric representation A S T Y 1, (2.7) B S T ΛY 1. (2.8) We are interested in selecting S so as to construct self-adjoint pencils. For matrix A to be symmetric, the matrix S must be such that S T Y 1 Y T S implying that the matrix Γ defined by Γ : Y T S T SY, (2.9) its inverse are symmetric. For matrix B to be symmetric, the matrix S must also be such that implying that S T ΛY 1 Y T Λ T S, Γ 1 Λ T ΛΓ 1. (2.10) Partition Γ 1 according to the sizes of sub-matrices in Λ, we will have: Γ 11 Γ 12 Γ 1l Γ 1k Γ 21 Γ 22 Γ 2l Γ 2k Γ 1, (2.11) Γ k 11 Γ k 12 Γ k 1l Γ k 1k Γ k1 Γ k2 Γ kl Γ kk where Γ ii is 2n i 2n i matrix, i 1, 2,..., l, Γ ii is n i n i matrix, i l+1,..., k. Substituting (2.11), (1.2) into (2.10), we obtain Γ i j (λ [2] j ) T λ [2] i Γ i j if 1 i, j l, Γ i j λ j λ [2] i Γ i j if 1 i l, l + 1 j k, (2.12) Γ i j λ j λ i Γ i j if l + 1 i, j k. First let us consider the case 1 i, j l. Without loss of generality, if we write U W Γ i j W V then (αi α j )U + (β i β j )W β i U + (α i α j )W β j V 0 0. β j U + (α i α j )W + β i V (β i β j )W (α i α j )V 0 0

5 Inverse Eigenvalue Problem with Non-simple Eigenvalues for Damped Vibration Systems 95 It is clear that if i j then Ui W i Γ ii, W i U i where U i W i are symmetric, give rise n i (n i + 1) degree of freedom.if i j, Γ i j 0. If 1 i l, l + 1 j k, then Γ i j is a 2 1 block matrix is equal to zero. Finally for l + 1 i, j k, if i j, then Γ i j 0, otherwise, Γ ii U i is arbitrary. Thus, we conclude that the symmetric matrix Γ 1 is equal to the following block-diagonal Γ 1 U1 W diag 1 Ul W,, l, U W 1 U 1 W l U l+1,, U k, (2.13) l where U i, i 1,, k, W i, i 1,, l are symmetric. Upon choosing an arbitrary Γ, a substitution by S T Y T Γ implies that the pencil λa B with A Y T ΓY 1, (2.14) B Y T ΓΛY 1 (2.15) is self-adjoint has eigenstructure (Λ, Y ). Moreover matrix A is invertible, its inverse matrix is 0 M A 1 1 M 1 M 1 C M 1. (2.16) We set A 1 Y Γ 1 Y T. (2.17) Substituting (2.16), (1.9) into (2.17), we have M ( Γ 1 Λ T T ) 1, (2.18) C M Γ 1 Λ T T M, (2.19) Γ 1 T 0. (2.20) Post-multiplying (1.8) by Γ 1 Λ T T applying (2.18) (2.19), we obtain K M 3 Γ 1 T M + C M 1 C. (2.21) We thus declare the following theorem. Theorem 2.2. Let a stard eigenpair (Λ, ) R 2n 2n R n 2n as (1.2), (1.4), be given then the general solution of QIEP forms are as: M ( Γ 1 Λ T T ) 1, C M Γ 1 Λ T T M, K M 3 Γ 1 T M + C M 1 C, where Γ 1 is given by (2.13), Γ 1 T 0. The following result can be regarded as the converse of Theorem 2.2.

6 96 M. Mohseni Moghadam A. Tajaddini Theorem 2.3. Let (Λ, ) R 2n 2n R n 2n be given matrices as in (1.2), (1.4). suppose there exists a symmetric nonsingular matrix Γ 1 R 2n 2n such that Γ 1 Λ T T is nonsingular, (2.20) (2.10) hold, then is nonsingular the eqnarray (1.8) holds for the self-adjoint quadratic pencil Q(λ) whose matrix coefficients M, C, K are defined according to (2.18), (2.19), (2.21), respectively. That is, (Λ, ) is a stard eigenpair for Q(λ). Proof. Since Γ 1 Λ T T is nonsingular, M can be defined. We want to show that in invertible. By substitution (2.16), (2.2) into (2.17), we have Γ 1 T Γ 1 T 0 M 1, (2.22) Γ 1 Λ T T M 1 Γ 1 Λ T T M 1 C M 1. (2.23) Hence (2.22) implies Γ 1 T M Also (2.23) implies 0. (2.24) I n Γ 1 Λ T T In M 1 M 1. (2.25) C post-multiplying (2.25) by M applying (2.24), we obtain (Γ 1 Λ T T M + Γ 1 T C) By (2.24), (2.26), we observe that Γ 1 Λ T T M + Γ 1 T C Γ 1 T M implies that the matrix is nonsingular. It follows that In. (2.26) 0 In 0, 0 I n 1 M 2 M 2 Γ 1 Λ T T M + Γ 1 T M Γ 1 T M K C, which is equivalent to M 2 + C + K 0.

7 Inverse Eigenvalue Problem with Non-simple Eigenvalues for Damped Vibration Systems Conclusion In this paper, we mainly present a general solution of an QIEP with a prescribed stard eigenpair (Λ, ) R 2n 2n R n 2n. Via appropriate choice of free variables in the general solution we can solve a quadratic pencil Q(λ) λ 2 M + λc + K with M, C K being symmetric.how to choice the total degree of freedoms in the general form of QIEP to obtain more exact quadratic pencil Q(λ). References [1] M.T. Chu, B.N. Datta, W.-W. Lin S.-F. u, The spill-over phenomenon in quadratic model updating, AIAA Journal 46 (2008), [2] Y.-C. Kuo, W.-W. Lin S.-F. u, On a general solution of partially described inverse quadratic eigenvalue problems its applications, NCTS Technical Report , National Tsing-Hua Univ., Taiwan. Mathematics/preprints/preprints/prep [3] M.T. Chu S-F. u, Spectral decomposition of self-adjoint quadratic pencils its applications, Math. Comput. 78 (2009), [4] T. Francoise M. Karl, The quadratic eigenvalue problem, SIAM Review 43 (2) (2001), [5] I. Gohberge, P. Lancaster L. Rodman, Matrix Polynomials, Academic Press, Inc., New York London, [6] D.J. Inman A.N. Andry, Jr., Some results on the nature of eigenvalues of discrete damped linear systems, Journal of Applied Mechanics 47(1980), [7] P. Lancaster, Inverse spectral problems for semi-simple problems damped vibrating systems, SIAM J. Matrix Anal. Appl. 29(2007), M. Mohseni Moghadam, Mahani Mathematical Research Center, Kerman , Iran. Mohseni@mail.uk.ac.ir A. Tajaddini, Department of Mathematics, Shahid Bahonar university of Kerman, Kerman , Iran. Azita_Tajaddinii@yahoo.com Received May 18, 2009 Accepted July 1, 2009