An Inverse Eigenvalue Problem of Hermite-hamilton Matrices in Structural Dynamic Model Updating

We first consider the following inverse eigenvalue problem: given X ∈ C n×m and a diagonal matrix Λ ∈ C m×m , find n×n Hermite-Hamilton matrices K and M such that KX MXΛ. We then consider an optimal approximation problem: given n × n Hermitian matrices K a and M a , find a solution K, M of the above inverse problem such that K − K a 2 M − M a 2 min. By using the Moore-Penrose generalized inverse and the singular value decompositions, the solvability conditions and the representations of the general solution for the first problem are derived. The expression of the solution to the second problem is presented.


Introduction
Throughout this paper, we will adopt the following notations.Let C m×n , HC n×n , and UC n×n stand for the set of all m × n matrices, n × n Hermitian matrices, and unitary matrices over the complex field C, respectively.By • we denote the Frobenius norm of a matrix.The symbols A T , A * , A −1 , and A † denote the transpose, conjugate transpose, inverse, and Moore-Penrose generalized inverse of A, respectively.We denote by HHC n×n the set of all n × n Hermite-Hamilton matrices.Vibrating structures such as bridges, highways, buildings, and automobiles are modeled using finite element techniques.These techniques generate structured matrix second-order differential equations: where M a , K a are analytical mass and stiffness matrices.It is well known that all solutions of the above differential equation can be obtained via the algebraic equation K a x λM a x.
But such finite element model is rarely available in practice, because its natural frequencies and mode shapes often do not match very well with experimentally measured ones obtained from a real-life vibration test 1 .It becomes necessary to update the original model to attain consistency with empirical results.The most common approach is to modify K a and M a to satisfy the dynamic equation with the measured model data.Let X ∈ C n×m be the measured model matrix and Λ diag δ 1 , δ 2 , . . ., δ m ∈ C m×m the measured natural frequencies matrix, where n ≥ m.The measured mode shapes and frequencies are assumed correct and have to satisfy where M, K ∈ C n×n are the mass and stiffness matrices to be corrected.To date, many techniques for model updating have been proposed.For undamped systems, various techniques have been discussed by Berman 2 and Wei 3 .Theory and computation of damped systems were proposed by authors of 4, 5 .Another line of thought is to update damping and stiffness matrices with symmetric low-rank correction 6 .The system matrices are adjusted globally in these methods.As model errors can be localized by using sensitivity analysis 7 , residual force approach 8 , least squares approach 9 , and assigned eigenstructure 10 , it is usual practice to adjust partial elements of the system matrices using measured response data.
The model updating problem can be regarded as a special case of the inverse eigenvalue problem which occurs in the design and modification of mass-spring systems and dynamic structures.The symmetric inverse eigenvalue problem and generalized inverse eigenvalue problem with submatrix constraint in structural dynamic model updating have been studied in 11 and 12 , respectively.Hamiltonian matrices usually arise in the analysis of dynamic structures 13 .However, the inverse eigenvalue problem for Hermite-Hamilton matrices has not been discussed.In this paper, we will consider the following inverse eigenvalue problem and an associated optimal approximation problem.Problem 1.Given that X ∈ C n×m and a diagonal matrix Λ ∈ C m×m , find n×n Hermite-Hamilton matrices K and M such that KX MXΛ.

1.3
Problem 2. Given that K a , M a ∈ HC n×n , let S E be the solution set of Problem 1. Find K, M ∈ S E such that We observe that, when M I, Problem 1 can be reduced to the following inverse eigenproblem: KX XΛ, 1.5 which has been solved for different classes of structured matrices.For example, Xie et al. considered the problem for the case of symmetric, antipersymmetric, antisymmetric, and persymmetric matrices in 14, 15 .Bai and Chan studied the problem for the case of centrosymmetric and centroskew matrices in 16 .Trench investigated the case of generalized symmetry or skew symmetry matrices for the problem in 17 and Yuan studied R-symmetric matrices for the problem in 18 .
The paper is organized as follows.In Section 2, using the Moore-Penrose generalized inverse and the singular value decompositions of matrices, we give explicit expressions of the solution for Problem 1.In Section 3, the expressions of the unique solution for Problem 2 are given and a numerical example is provided.

Solution of Problem 1
where U is the same as in 2.1 .

Proof. Let
Let the partition of the matrix U * X be where U is defined as in 2.1 .We assume that the singular value decompositions of the matrices X 1 and X 2 are Let the singular value decompositions of the matrices X 2 ΛV 2 and X 1 ΛS 2 be where Theorem 2.3.Suppose that X ∈ C n×m and Λ ∈ C m×m is a diagonal matrix.Let the partition of U * X be 2.3 , and let the singular value decompositions of X 1 , X 2 , X 2 ΛV 2 , and X 1 ΛS 2 be given in 2.4 and 2.5 , respectively.Then 1.3 is solvable and its general solution can be expressed as where

2.8
Using 2.3 , the above equation is equivalent to the following two equations: By the singular value decomposition of X 2 , then the relation 2.9 becomes 11 Clearly, 2.11 with respect to unknown matrix F is always solvable.By Lemma 2.2 and 2.5 , we get where L ∈ C k× k−t is an arbitrary matrix.Substituting F LP * 2 into 2.12 , we get

2.14
Since W 1 is of full column rank, then the above equation with respect to unknown matrix N is always solvable, and the general solution can be expressed as

2.15
where G ∈ C k× k−s is an arbitrary matrix.Substituting F LP * 2 and 2.15 into 2.10 , we get

2.16
By the singular value decomposition of X 1 , then the relation 2.16 becomes 2.17

2.18
Clearly, 2.17 with respect to unknown matrix L is always solvable.From Lemma 2.2 and 2.5 , we have where J ∈ C k−g × k−t is arbitrary.Substituting L T 2 J into 2.18 , we get

2.20
Then, we have

6 Mathematical Problems in Engineering
Since R * 1 is of full row rank, then the above equation with respect to GW * 2 is always solvable.By Lemma 2.2, we have where Y 1 ∈ C k×k is arbitrary.Then, we get

2.23
where Finally, we have From Lemma 2.1, we have that if the mass matrix M ∈ HHC n×n , then M is not positive definite.If M is symmetric positive definite and K is a symmetric matrix, then 1.3 can be reformulated as the following form: AX XΛ,

2.25
where A M −1 K. From 20, Theorem 7.6.3, we know that A is a diagonalizable matrix, all of whose eigenvalues are real.Thus, Λ ∈ R m×m and X is of full column rank.Assume that X is a real n × m matrix.Let the singular value decomposition of X be where OR n×n denotes the set of all orthogonal matrices.The solution of 2.25 can be expressed as where Z 12 ∈ R m× n−m is an arbitrary matrix and Z 22 ∈ R n−m × n−m is an arbitrary diagonalizable matrix see 21, Theorem 3.1 .
, where G ∈ R n−m × n−m is an arbitrary nonsingular matrix and Λ 2 diag λ q 1 I k q 1 , . . ., λ p I k p with λ p > • • • > λ q 1 > λ q .The solutions to 1.3 with respect to unknown matrices M > 0 and K K T are presented in the following theorem.Theorem 2.4 see 21 .Given that X ∈ R n×m , rank X m, and Λ diag λ 1 I k 1 , . . ., λ q I k q ∈ R m×m , let the singular value decomposition of X be 2.26 .Then the symmetric positive-definite solution M and symmetric solution K to 1.3 can be expressed as

Solution of Problem 2
Lemma 3.

3.1
Then Z ∈ S a if and only if Z ∈ S b .
For the given matrices K a , M a ∈ HC n×n , let

3.2
From Theorem 2.3, we know that S E / ∅.The following theorem is for the best approximation solution of Problem 2. Theorem 3.2.Given that X ∈ C n×m , Λ ∈ C m×m , and K a , M a ∈ HC n×n , then Problem 2 has a unique solution and the solution can be expressed as where

3.4
Proof.It is easy to verify that S E is a closed convex subset of HHC n×n × HHC n×n .From the best approximation theorem, we know that there exists a unique solution K, M in S E such that 1.4 holds.From Theorem 2.3 and the unitary invariant of the Frobenius norm, we have

3.7
Then from the unitary invariant of the Frobenius norm, we have we have h 0. In other words, we can always find Y such that h 0. Let

3.10
Then, we have that f min is equivalent to g min.According to Lemma 3.1 and 3.10 , we get the following matrix equation: we have that, when F F, g attains its minimum, which gives Y R Then, the unique solution of Problem 2 given by 3.3 is obtained.Now, we give an algorithm to compute the optimal approximate solution of Problem 2.

Algorithm.
1 Input K a , M a , X, Λ, and U.

Definition 1 . 1 .
Let J n 0 I k −I k 0 , n 2k, and A ∈ C n×n .If A A * and J n AJ n A * , then the matrix A is called Hermite-Hamilton matrix.