Symmetric inverse generalized eigenvalue problem with submatrix constraints in structural dynamic model updating

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1 Electronic Journal of Linear Algebra Volume Volume 011 Article Symmetric inverse generalized eigenvalue problem with submatrix constraints in structural dynamic model updating Meixiang Zhao Zhigang Jia Musheng Wei Follow this and additional works at: Recommended Citation Zhao, Meixiang; Jia, Zhigang; and Wei, Musheng. 011, "Symmetric inverse generalized eigenvalue problem with submatrix constraints in structural dynamic model updating", Electronic Journal of Linear Algebra, Volume. DOI: his Article is brought to you for free and open access by Wyoming Scholars Repository. It has been accepted for inclusion in Electronic Journal of Linear Algebra by an authorized editor of Wyoming Scholars Repository. For more information, please contact

2 Volume, pp , August 011 SYMMERIC INVERSE GENERALIZED EIGENVALUE PROBLEM WIH SUBMARIX CONSRAINS IN SRUCURAL DYNAMIC MODEL UPDAING MEIXIANG ZHAO, ZHIGANG JIA, AND MUSHENG WEI Abstract. In this literature, the symmetric inverse generalized eigenvalue problem with submatrix constraints and its corresponding optimal approximation problem are studied. A necessary and sufficient condition for solvability is derived, and when solvable, the general solutions are presented. Key words. Symmetric inverse generalized eigenvalue problem, Submatrix constraint, Optimal approximation. AMS subject classifications. 15A15, 15A9. 1. Introduction. Let R m n denote the set of all m n real matrices, SR n n the set of all n n real symmetric matrices, and C n the n-dimensional complex vector space. he dynamic analysis of a mechanical or civil structure by the finite element technique can be modelled by a generalized eigenvalue problem 1.1 K a x = λm a x, where K a,m a SR n n are the analytical stiffness matrix and mass matrix, respectively, and λ C and x C n are the generalized eigenvalue and corresponding generalized eigenvector. It s well known that the natural frequencies and mode shapes of such a finite element model and ones experimentally measured by a vibration test do not match very well [4], [0]. he finite element model updating problem is to determine how to update the model to closely match the experimental model data that gives an incomplete set of eigenpairs. Let x 1,...,x p C n be the measured modal vectors and λ 1,...,λ p C be the Received by the editors on July 30, 010. Accepted for publication on June 7, 011. Handling Editor: Bryan Shader. Kewen Institute, Xuzhou Normal University, Jiangsu 1116, P.R. China. Supported by the Natural Science Foundation of Shandong Province under grant ZR010AM014. Department of Mathematics, Xuzhou Normal University, Jiangsu 1116, P.R. China zhgjia@xznu.edu.cn. Supported by the Universities Natural Science Foundation of Jiangsu Province under grant 10KJD110005, the NSFC under grant , and the Natural Science Foundation of Jiangsu Province under grant BK College of Mathematics and Science, Shanghai Normal University, Scientific Computing Key Laboratory of Shanghai Universities, Shanghai 0034, P.R. China. Supported by the NSFC under grant , and the Shanghai Leading Academic Discipline Project under grant S

3 Volume, pp , August M. Zhao, Z. Jia, and M. Wei measured natural frequencies, where p n. he measured mode shapes and frequencies are assumed to be correct, and the mass matrix M SR n n and stiffness matrix K SR n n to be constructed have to satisfy the dynamic equation 1. Kx i = Mx i λ i, i = 1,...,p see e.g. [1], [], [6] [8], [11], [1], [16], [17], []. Since we are only interested in real matrices, the prescribed eigenpairs must be closed under complex conjugation. Without causing ambiguity, the p prescribed eigenpairs are represented by the matrix form X,Λ, where each block of Λ R p p is either a 1-by-1 matrix or a -by- matrix whose eigenvalues are a complex conjugate pair, and X R n p represents the eigenvector matrix in the sense that each pair of column vectors associated with a -by- block in Λ retains the real and the imaginary part, respectively, of the original complex eigenvector. hen 1. is equivalent to KX = MXΛ. As model errors can be localized by sensitivity analysis [10], [19], residual force approach [9], least-squares approach [13] and assigned eigenstructure [5], it is a usual practice to adjust partial elements of the analytical mass and stiffness matrices. Mathematically, such a partially updating problem can be described as the following two problems: Problem 1.1. Given p eigenpairs X,Λ, where X R n p and Λ R p p are described as the above and K 0, M 0 SR r r, 1 p, r n, p + r n, find real matrices K, M SR n n such that 1.3 KX = MXΛ, K[1,r] = K 0, M[1,r] = M 0, where K[1,r] and M[1,r] are the r r leading principal submatrices of K and M, respectively. Problem 1.. Given K a,m a SR n n with K a [1,r] = K 0,M a [1,r] = M 0, find ˆK, ˆM S E such that 1.4 K a,m a K,M = inf K a,m a ˆK, ˆM, ˆK, ˆM S E where S E is the solution set of Problem 1.1. Here denotes the Frobenius norm. he second problem is to find the best approximation for a given symmetric matrix pencil under a given spectral constraint and a symmetric submatrix pencil constraint. Such a problem always arises in structural dynamic model updating. Y. Yuan and H. Dai [1] solved the above two problems, where K a, K 0, K, M a, M 0, and M are free from the symmetry constraint conditions. In this paper, we consider the symmetric cases in the dynamic analysis of a mechanical or civil structure by the finite element technique. Only by using the Moore-Penrose generalized inverse and the singular value decomposition, we present the solvability condition and the

4 Volume, pp , August 011 Symmetric Inverse Generalized Eigenvalue Problem With Submatrix Constraints 731 expression for the solutions of Problems 1.1 and 1.. For a special case, we derive a formula of the general solution which is inexpensive to compute and can be used routinely in practice. hroughout this paper On stands for the set of all n n orthogonal matrices; A the transpose of a real matrix A; A the Moore-Penrose generalized inverse of A; P A = AA, P A = I P A; tra the trace of the matrix A R n n ; and ranka the rank of A. For A,B R m n, A,B = trb A denotes an inner product in R m n.. General solutions of Problems 1.1 and 1.. o facilitate the following discussion, we first make some notations. Let X R n p, K, M SR n n have the following partitions:.1 X = X1 X K0 K, K = 1 K1 K M0 M, M = 1 M1 M where X 1 and X have r and n r rows respectively, K 0,M 0 R r r, K 1,M 1 R r n r, and K,M SR n r n r. hen KX = MXΛ can be rewritten as K0 K 1 X1 M0 M K1 = 1 X1. K X M1 Λ. M X hat implies that 1.3 is equivalent to,.3.4 K 1 X = M 0 X 1 Λ K 0 X 1 + M 1 X Λ, K 1 X 1 = M 1 X 1 Λ K X + M X Λ. Problem 1.1 can be solved by computing the solutions K 1, M 1, K and M of.3 and.4. Suppose that the singular value decomposition of X has been computed Σ 0.5 X = U V, 0 0 where U = [U 1,U ] On r, V = [V 1,V ] Op, Σ = diagσ 1,...,σ s, σ i > 0, i = 1,...,s, s = rankx, U 1 R n r s, V 1 R p s. Similarly, suppose that the following singular value decompositions have been computed Ω 0.6 X ΛV = [P 1,P ] [Q 1,Q ], Ω X 1 V Q = [ P 1, P 0 ] 0 0 ˆΣ X ΛX 0 = [Û1,Û] 0 0 [ Q 1, Q ], [ˆV 1, ˆV ].

5 Volume, pp , August M. Zhao, Z. Jia, and M. Wei Now we recall some known results, which will be used repeatedly in this paper. Lemma.1. [14] Let Y,B R m k,m k, be given. hen AY = B has a symmetric solution A SR m m if and only if BP Y = B and P Y BY = P Y BY. Moreover, the solution set is S = {BY + BY P Y + P Y HP Y : H SR m m }, and the minimal solution under the Frobenius norm is A opt = BY + BY P Y. Lemma.. [3] If E R m n,f R p q and G R m q,.9 EXF = G has a solution X R p q if and only if EE GF F = G, in which case, the general solution of.9 can be expressed as where Y R n p is an arbitrary matrix. X = E GF + Y E EY FF, he next lemma is extracted from the proof of heorem 1 in [1]. Lemma.3. [1] Equation K 0 X 1 + K 1 X = M 0 X 1 Λ + M 1 X Λ with respect to matrices K 1 and M 1 is solvable if and only if.10 K 0 X 1 V Q = M 0 X 1 ΛV Q, in which case, we have.11.1 K 1 = K 10 + LP X ΛX + WU, M 1 = M 10 + LP, where M 10 = K 0 X 1 V M 0 X 1 ΛV X ΛV, K 10 = [M 0 X 1 Λ K 0 X 1 + M 10 X Λ]X, and L R r n r t, W R r n r s are arbitrary matrices.

6 Volume, pp , August 011 Symmetric Inverse Generalized Eigenvalue Problem With Submatrix Constraints 733 As we discussed above, the general solution of Problem 1.1 can be derived by solving.3 and.4. Lemma.3 has presented the solvability condition and the general solution if exists of.3. Now we study.4. Define L = P L, W = U W, and Ψ = M 10X 1 Λ K 10X 1 + LX 1 Λ X ΛX LX1. Substitute.11 and.1 into.4, to get.13 K X = Ψ WX 1 + M X Λ with respect to unknown real symmetric matrices K and M. Repeatedly applying Lemma.1 and equations.5.8, and.13 has a symmetric solution K if and only if.14 M X ΛV = [ Ψ + WX 1 ]V,.15 U 1 U 1 [Ψ + M X Λ]X SRn r n r. Equation.14 has a symmetric solution M if and only if.16 X ΛX LX1 V Q = K 10 + WX 1 V Q,.17 P 1 P 1 [ Ψ + WX 1 ]V X 1 ΛV SR r r. Now we prove that.16 always holds. Lemma. will be repeatedly used without being mentioned in the following analysis. Equation.16 has a solution L if and only if.18 [I X ΛX X ΛX ]K 10 + WX 1 V Q = 0. Recalling the SVD of X ΛX as in.8,.18 can be rewritten as.19 Û WX 1 V Q = Û K 10X 1 V Q. We can see that.19 is always solvable and the expression of its general solution is.0 W = ÛÛ K 10X 1 V Q X 1 V Q + Ŷ ÛÛ Ŷ X 1V Q X 1 V Q, where Ŷ Rn r r is arbitrary. hat means.18 always holds and consequently there must exist a matrix L satisfying.16. Now substitute.0 into.16, to get.1 X ΛX LX1 V Q = Û1Û 1 K 10 Ŷ X 1V Q.

7 Volume, pp , August M. Zhao, Z. Jia, and M. Wei he general solution of.1 has the form. L = X ΛX K 10 Ŷ P 1 P 1 + Ỹ ˆV 1 ˆV 1 Ỹ P 1 P 1, where Ỹ is arbitrary. So that.16 is always solvable and its general solution can be expressed as in.. By Lemma.1, we can conclude that.14 is solvable when.16 and.17 hold, and its general solution is.3m = B M X ΛV + B M X ΛV P X ΛV + P X ΛV H M P X ΛV in which B M = [ Ψ + X ΛX LX1 + WX 1 ]V and H M SR n r n r is arbitrary. Similarly, if.14 and.15 hold, then.13 has a symmetric solution.4 K = B K X + B K X P X + P X H K P X, where B K = Ψ X ΛX LX1 WX 1 + M X Λ and H K SR n r n r is arbitrary. Now we suppose that there exist Ŷ,Ỹ and H M such that the symmetric conditions.15 and.17 hold after substituting.0,., and.3 into them. hen we can conclude that Problem 1.1 is solvable from the above analysis. heorem.4. Problem 1.1 is solvable if and only if.10,.15 and.17 hold. In this case, the general solution is K0 K K,M = 1 M 0 M 1 K1 K M1, M where K 1, M 1, M, K are given by.11,.1,.3, and.4 in which Ŷ, Ỹ, and H M satisfy.15 and.17. Now we study Problem 1. under the condition that the solution set S E of Problem 1.1 is not empty. Firstly, recall an useful result in [15]. Lemma.5. [15] Suppose that A R q m, R q q, and Γ R m m, where = = and Γ = Γ = Γ. hen A DΓ = if and only if A DΓ = 0, in which case, min A EΓ E R q m A DΓ = A AΓ.

8 Volume, pp , August Symmetric Inverse Generalized Eigenvalue Problem With Submatrix Constraints 735 For given matrices K a R n n and M a R n n in Problem 1., define K a = K 0 K a 1 K a 1 K a, M a = M 0 M a 1 M a 1 M a where K a 1,Ma 1 R r n r, and K a,ma SR n r n r. Given two matrices C, D and a matrix set S, the notation C A + D B = min means that C A + D B is minimized by A,B S; i.e., C A + D B = min Â, ˆB S C Â + D ˆB., heorem.6. Under the conditions of heorem.4 and assuming that Ŷ,Ỹ and H M are unique solutions of.15 and.17, Problem 1. has a unique solution K0 K K,M = 1 M 0 M 1 K1 K M1 M, where K 1, M 1, M, K are given by.11,.1,.3, and.4, and H K and H M satisfy PX K a H K PX = 0 and PX ΛV Ma H M PX = 0, ΛV respectively. Proof. Equation 1.4 is equivalent to.6 K a K + M a M = min. With the partitions.1 and.5 in mind, we have K a K + M a M = K a 1 K 1 + K a K + M a 1 M 1 + M a M. hen.6 holds if and only if.7 K a K + M a M = min. By Lemma.5,.7 holds if and only if U U K a H K U U = 0, P P M a H M P P = 0. Remark.7. We have solved Problem 1.1 and Problem 1. under the assumption that.15 and.17 hold for some Ŷ, Ỹ, and H M. In general to find such Ŷ, Ỹ, and H M, we have to solve two matrix equations with three unknowns after substituting.0,., and.3 into.15 and.17.

9 Volume, pp , August M. Zhao, Z. Jia, and M. Wei o end this section, we consider a sufficient condition that.8 K 0 X 1 = M 0 X 1 Λ, K 1 = 0, M 1 = 0, under which. is reduced to K0 0 0 K X1 X M0 0 = 0 M X1 X Λ; i.e.,.9 K X = M X Λ. o solve.9 for symmetric matrices K and M is the symmetric inverse eigenvalue problem without constraints. Dai [7] studied this problem and gave an efficient algorithm for it by using QR-like decomposition with column pivoting and least squares techniques. Here we present the expression of the general symmetric solution only by generalized inverse and SVD. heorem.8. Suppose X and Λ are known and the singular value decomposition of X is.5. hen.9 has a symmetric solution and K, M = U M 11 ΣΛ 11 Σ 1 M 1 ΣΛ 11 Σ 1 M 11 M 1 M 1 ΣΛ 11 Σ 1 K M 1 M diagu, U, where M 11 = P Y HP Y with H being a symmetric matrix such that P Y HP Y ΣΛ 11 Σ 1 is symmetric, Y = ΣΛ 1, M 1 = FP Y with F being arbitrary, K,M are two arbitrary symmetric matrices, Λ 11 = V 1 ΛV 1, Λ 1 = V 1 ΛV. Proof. Substitute.5 into.9. Before we consider Problem 1. under the condition.8, we give some notation. K a 11 K a 1 M a 11 M a 1 K a 1 K a M a 1 M a = U K a M a diagu,u, Λ 11 = ΣΛ 11 Σ 1 I, KM11 = K a 11 M a 11, KM1 = K a 1 M a 1. heorem.9. If K 0 X 1 = M 0 X 1 Λ and K 1 = 0, M 1 = 0, Problem 1. has a solution and M K, M = U 11 ΣΛ 11 Σ 1 M 1 ΣΛ 11 Σ 1 M 11 FPY M 1 ΣΛ 11 Σ 1 K a FPY M a diagu, U,

10 Volume, pp , August 011 Symmetric Inverse Generalized Eigenvalue Problem With Submatrix Constraints 737 in which M 11 = PY HP Y with H being a symmetric solution of P Y HP Y Λ 11 KM 11 = min; PY HP Y ΣΛ 11 Σ 1 = Σ 1 Λ 11 ΣPY HP Y ; and F = KM 1 PY Λ 11 +WP PY Λ with W being arbitrary. 11 Proof. It is clear that K K a + M M a = min if and only if.30 K K a F + M M a F = min. Equation.30 holds if and only if M Λ KM 11 = min, M Λ 1 11 KM 1 = min, K Ka F = min, M Ma F = min. Remark.10. For details about solving heorem 3.3. in [18]. min H SR n n P Y HP Y Λ 11 KM 11, see 3. A special case. In this section, we consider one simple but not easy case of Problem 1.1 and Problem 1.. Problem 3.1. Given p eigenpairs by X,Λ, where X R n p and Λ R p p is a block diagonal matrix with -by- blocks or single real eigenvalues on its main diagonal, and K 0 SR r r. Find a real matrix K SR n n such that 3.1 KX = XΛ, K[1,r] = K 0, where K[1,r] is the r r leading principal submatrix of K. Problem 3.. Given K a SR n n with K a [1,r] = K 0, find ˆK S E such that K a ˆK = inf K S E K a K, where S E is the solution set of Problem 3.1. Let X R n p and K SR n n have the partitions as in.1 and the singular value decomposition of X be given as in.5. heorem 3.3. Suppose that K 0 SR r r, X R n p, and Λ R p p are given as in Problem 3.1. hen Problem 3.1 is solvable if and only if 3. and 3.3 hold. K 0 X 1 X 1 ΛP X = 0, P X Λ X X1 X 1 Λ K 0 X 1 X = 0, X XΛ = Λ X X

11 Volume, pp , August M. Zhao, Z. Jia, and M. Wei If 3. and 3.3 hold, then the general solution of Problem 3.1 is 3.4 K = K 0 X 1 Λ K 0 X 1 X + P X 1P X LP X K1 BX + BX PX + PX HPX, where B = X Λ X X 1 Λ K 0 X 1 X 1 PX L P X 1P X X 1, both L R r n r and H SR n r n r are arbitrary. Proof. Applying.1, 3.1 is equivalent to the following two equations K 1 X = X 1 Λ K 0 X 1, K 1 X 1 = X Λ K X. By Lemma., 3.5 has a solution K 1 if and only if X 1 Λ K 0 X 1 P X = 0, in which case, 3.7 K 1 = X 1 Λ K 0 X 1 X + Y P X, where Y R r n r is arbitrary. Substitute 3.7 into 3.6, 3.8 K X = X Λ K 1 X 1 = X Λ X X 1 Λ K 0 X 1 X 1 P X Y X 1. By Lemma.1, there exists a symmetric matrix K satisfying 3.8 if and only if X Λ K 1 X 1 P X = 0, P X X Λ K 1 X 1 X SRq n r. Let G = P Λ X X X1 X 1 Λ K 0 X 1 X. Since P P X X X1 P X X1 = 0, 3.11 P X X 1 P X X 1 GP X P X 0. Substitute 3.7 into 3.9, to get 3.1 P X X 1 Y P X = G. By 3.11 and Lemma., 3.1 holds if and only if G = 0, which is exactly the second equation in 3., and in this case 3.13 Y = L P X X 1 P X X 1 LPX, where L R r n r is arbitrary. Substituting 3.13 into 3.7, we have 3.14 K 1 = X 1 Λ K 0 X 1 X + P X 1P X LP X.

12 Volume, pp , August 011 Symmetric Inverse Generalized Eigenvalue Problem With Submatrix Constraints 739 Now we substitute 3.14 into 3.10, to get P X X Λ K 1 X 1 X = X Λ X X K 1 X 1 X = X ΛX X Λ X 1 X 1 X + X 1X K 0 X 1 X = X X X Λ Λ X 1 X 1 X + X 1X K 0 X 1 X. So we can see that the condition 3.10 is equivalent to 3.3. At this point, we have found the necessary and sufficient condition for the solvability of Problem 3.1. Next we prove that 3.4 is the general solution of Problem 3.1 if 3. and 3.3 hold. If 3. and 3.3 hold, we have proved that K 1 has the expression as in 3.14; what is left to prove is K = BX +BX P X +P X HP X. Indeed, if 3. and 3.3 hold, then 3.9 and 3.10 are satisfied. Applying Lemma.1 and substituting 3.13 into 3.8, 3.8 has a symmetric solution K = BX + BX P X + P X HP X. If X is square and nonsingular, then PX = 0 and P X X 1 X 1 Λ K 0 X 1 X 1. So we have = 0 and B = X Λ Corollary 3.4. Suppose that K 0 SR r r, X R n p, and Λ R p p is a block diagonal matrix. If X is square and nonsingular, then Problem 3.1 has a unique solution K = K0 X 1 Λ K 0 X 1 X 1 K 1 if and only if X XΛ = ΛX X. X ΛX 1 X 1 X 1 Λ K 0 X 1 X 1 X 1 Now we present a general solution of Problem 3. assuming Problem 3.1 is solvable. heorem 3.5. For given matrices K a R n n with K a [1,r] = K 0 as in.5. If 3. and 3.3 hold, then Problem 3. has a unique solution K 0 X 1 Λ K 0 X 1 X ˆK = + L K1 BX + BX PX + PX K a P X, where B = X ΛX X X 1 Λ K 0 X 1 X 1 X L X 1 X, L satisfies 3.15 vec L = [X 1 X X 1 X Q + I I + X 1X X 1X ] 1 vec K 1 X 1 X K + K, Q is a permutation matrix such that vec L = Qvec L and K 1 = K a 1 X 1 Λ K 0 X 1 X. Proof. Let L = P X 1P X LP X. From heorem 3.3, K a K = K a 1 K 1 + K a K

13 Volume, pp , August M. Zhao, Z. Jia, and M. Wei By Lemma.5, = K a 1 X 1 Λ K 0 X 1 X L + K a X ΛX + X X 1 Λ K 0 X 1 X 1 X + L X 1 X + X 1X L P X HP X. min K a K = K a H=H 1 X 1 Λ K 0 X 1 X L + K a X ΛX + X X 1 Λ K 0 X 1 X 1 X + L X 1 X + X 1X L P X K a P X, and the minimum is attained if and only if PX H K a P X = 0. Let K 1 = K a 1 X 1 Λ K 0 X 1 X, K = K a PX K a P X X ΛX +X X 1 Λ K 0 X 1 X 1 X, and hen f L = K 1 L + K + L X 1 X + X 1X L. f L = tr K 1 K1 + tr L L + tr K K + tr K X1 X L +tr K X 1 X L + trx1 X X 1X L L +trx 1 X LX1 X L 4tr K1 L. Consequently, to get the minimal value of f L, we should find its minimal point. As f L L = X 1X L X 1 X + 4I + X 1X X 1X L 4 K 1 + X 1 X K + K, f L L = 0 is equivalent to 3.16 Φvec L = vec K 1 X 1 X K + K, where Φ = X 1 X X 1 X Q+I I+X 1X X 1X, and Q is a permutation matrix such that vec L = Qvec L. Since Φ is nonsingular, 3.16 has an unique solution 3.15 and f L has an unique minimal value point L. 4. Conclusions. Motivated by Y. Yuan and H. Dai [1], this paper is concerned with the symmetric inverse generalized eigenvalue problem and the problem of the optimal approximation for a given symmetric matrix pencil under a given spectral constraint and a symmetric submatrix pencil constraint. By using the Moore-Penrose generalized inverse and the singular value decomposition, we first present the solvability condition and the expression for the solution of these two problems. For the

14 Volume, pp , August 011 Symmetric Inverse Generalized Eigenvalue Problem With Submatrix Constraints 741 case that the mass matrix is unit, there is a formula of the general solution which is inexpensive to compute and can be used routinely in practice. Acknowledgment. he authors are grateful to the editor and the anonymous referee for their useful comments and suggestions, which greatly improved the original presentation. REFERENCES [1] M. Baruch. Optimization procedure to correct stiffness and flexibility matrices using cibration tests. AIAA Journal, 16: , [] M. Baruch. Optimal correction of mass and stiffness matrices using measured modes. AIAA Journal, 0: , 198. [3] A. Ben-Israel and.n.e. Greville. Generalized Inverses: heory and Applications, nd edition. Springer-Verlag, New York, 003. [4] A. Berman and W.G. Flannely. heory of incomplete models of dynamic structures. AIAA Journal, 9: , [5] R.G. Cobb and B.S. Liebst. Structural damage identification using assigned partial eigenstructure. AIAA Journal, 35:15 158, [6] H. Dai. About an inverse eigenvalue problem arising in vibration analysis. RAIRO Modélisation Mathématique et Analyse Numérique 9:41 434, [7] H. Dai. An algorithm for symmetric generalized inverse eigenvalue problems. Linear Algebra and its Applications, 96:79 98, [8] W.L. Li. A new method for structural model updating and joint stiffness identification. Mechanical Systems and Signal Processing, 16: , 00. [9] M. Link. Identification and correction of errors in analytical models using test data-theoretical and practical bounds. he Proceedings of the 8th International Modal Analysis Conference, Florida, USA, , [10] W. Luber and A. Lotze. Application of sensitivity methods for error localization in finite element systems. he Proceedings of the 8th International Modal Analysis Conference, Florida, USA, , [11] C. Minas and D.J. Inman. Matching finite element models to modal data. Journal of Vibration and Acoustics, 11:84 9, [1] J.E. Mottershead and M.I. Friswell. Model updating in structural dynamics: a survey. Journal of Sound and Vibration, 167: , [13] J.C. O Callahan and C.M. Chou. Localization of model errors in optimized mass and stiffness matrices using modal-test data. International Journal of Analytical and Experimental Analysis, 4:8 14, [14] F. isseur. A chart of backward errors for singly and doubly structured eigenvalue problems. SIAM Journal on Matrix Analysis and Applications, 4: , 003. [15] W.F. rench. Inverse eigenproblems and associated approximation problems for matrices with generalized symmetry or skew symmetry. Linear Algebra and its Applications, 380:199 11, 004. [16] F.S. Wei. Mass and stiffness interaction effects in analytical model modification. AIAA Journal, 8: , [17] F.S. Wei and D.W. Zhang. Mass matrix modification using element correction method. AIAA Journal, 7:119 11, [18] M. Wei. heory and Calculation of Generalized Least Square Problem. Science Press, Beijing,

15 Volume, pp , August M. Zhao, Z. Jia, and M. Wei 006 in Chinese. [19] H.Q. Xie. Sensitivity Analysis of Eigenvalue Problem. Ph.D. Dissertation, Nanjin University of Aeronautics and Astronautica, 003. [0] Y. Yuan and H. Dai. Inverse problems for symmetric matrices with a submatrix constraint. Applied Numerical Mathematics, 57: , 007. [1] Y. Yuan and H. Dai. A generalized inverse eigenvalue problem in structural dynamic model updating. Journal of Computational and Applied Mathematics, 6:4 49, 009. [] O. Zhang, A. Zerva, and D.W. Zhang. Stiffness matrix adjustment using incomplete measured modes. AIAA Journal, 35: , 1997.