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

Abstract

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.

1. 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.

Definition 1.1. Let 𝐽_𝑛=(^0𝐼𝑘_−𝐼𝑘0), 𝑛=2𝑘, and 𝐴∈𝐶^𝑛×𝑛. If 𝐴=𝐴^∗ and 𝐽_𝑛𝐴𝐽_𝑛=𝐴^∗, then the matrix 𝐴 is called Hermite-Hamilton matrix.

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:𝑀_𝑎𝑧(̈𝑡)=𝐾_𝑎𝑧(𝑡), (1.1) 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_𝑎𝑥=𝜆𝑀_𝑎𝑥. 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(𝛿₁,𝛿₂,…,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 𝐾𝑋=𝑀𝑋Λ, (1.2) 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 ‖𝐾−𝐾_𝑎‖²+‖𝑀−𝑀_𝑎‖²=min_{(𝐾,𝑀)∈𝑆𝐸}(‖𝐾−𝐾_𝑎‖²+‖𝑀−𝑀_𝑎‖²). (1.4)

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.

2. Solution of Problem 1

Let1𝑈=√2𝐼_𝑘𝐼_𝑘−𝑖𝐼_𝑘𝑖𝐼_𝑘. (2.1)

Lemma 2.1. Let 𝐴∈𝐶^𝑛×𝑛. Then 𝐴∈𝐻𝐻𝐶^𝑛×𝑛 if and only if there exists a matrix 𝑁∈𝐶^𝑘×𝑘 such that 𝑁𝐴=𝑈0𝑁_∗0𝑈^∗, (2.2) where 𝑈 is the same as in (2.1).

Proof. Let 𝐴=(^𝐴11𝐴12_𝐴^∗_12𝐴22), and let each block of 𝐴 be square. From Definition 1.1 and (2.1), it can be easily proved.

Lemma 2.2 (see [19]). Let 𝐴∈𝐶^𝑚×𝑛, 𝐵∈𝐶^𝑝×𝑞, and 𝐸∈𝐶^𝑚×𝑞. Then the matrix equation 𝐴𝑋𝐵=𝐸 has a solution 𝑋∈𝐶^𝑛×𝑝 if and only if 𝐴𝐴^†𝐸𝐵^†𝐵=𝐸; in this case the general solution of the equation can be expressed as 𝑋=𝐴^†𝐸𝐵^†+𝑌−𝐴^†𝐴𝑌𝐵𝐵^†, where 𝑌∈𝐶^𝑛×𝑝 is arbitrary.

Let the partition of the matrix 𝑈^∗𝑋 be𝑈^∗𝑋𝑋=¹𝑋₂,𝑋₁,𝑋₂∈𝐶^𝑘×𝑚, (2.3) where 𝑈 is defined as in (2.1).

We assume that the singular value decompositions of the matrices 𝑋₁ and 𝑋₂ are𝑋₁=𝑅𝑆𝐷000^∗,𝑋₂=𝑊𝑉Σ000^∗, (2.4) where 𝑅=(𝑅₁,𝑅₂)∈𝑈𝐶^𝑘×𝑘, 𝑆=(𝑆₁,𝑆₂)∈𝑈𝐶^𝑚×𝑚, 𝐷=diag(𝑑₁,…,𝑑_𝑙)>0,𝑙=rank(𝑋₁), 𝑅₁∈𝐶^𝑘×𝑙,𝑆₁∈𝐶^𝑚×𝑙, and 𝑊=(𝑊₁,𝑊₂)∈𝑈𝐶^𝑘×𝑘, 𝑉=(𝑉₁,𝑉₂)∈𝑈𝐶^𝑚×𝑚, Σ=diag(𝜎₁,…,𝜎_𝑠)>0,𝑠=rank(𝑋₂), 𝑊₁∈𝐶^𝑘×𝑠,𝑉₁∈𝐶^𝑚×𝑠.

Let the singular value decompositions of the matrices 𝑋₂Λ𝑉₂ and 𝑋₁Λ𝑆₂ be𝑋₂Λ𝑉₂=𝑃𝑄Ω000^∗,𝑋₁Λ𝑆₂=𝑇𝐻Δ000^∗, (2.5) where 𝑃=(𝑃₁,𝑃₂)∈𝑈𝐶^𝑘×𝑘, 𝑄=(𝑄₁,𝑄₂)∈𝑈𝐶^{(𝑚−𝑠)×(𝑚−𝑠)}, Ω=diag(𝜔₁,…,𝜔_𝑡)>0,𝑡=rank(𝑋₂Λ𝑉₂), 𝑃₁∈𝐶^𝑘×𝑡,𝑄₁∈𝐶^{(𝑚−𝑠)×𝑡}, and 𝑇=(𝑇₁,𝑇₂)∈𝑈𝐶^𝑘×𝑘, 𝐻∈𝑈𝐶^{(𝑚−𝑙)×(𝑚−𝑙)}, Δ=diag(𝑎₁,…,𝑎_𝑔)>0,𝑔=rank(𝑋₁Λ𝑆₂),  𝑇₁∈𝐶^𝑘×𝑔.

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 𝑋₁,𝑋₂,𝑋₂Λ𝑉₂, and 𝑋₁Λ𝑆₂ be given in (2.4) and (2.5), respectively. Then (1.3) is solvable and its general solution can be expressed as 𝐹𝑀=𝑈0𝐹_∗0𝑈^∗,𝐾=𝑈0𝐹𝑋₂Λ𝑋^†₂+𝐺𝑊^∗₂(𝐹𝑋₂Λ𝑋^†₂+𝐺𝑊^∗₂)^∗0𝑈^∗, (2.6) where 𝐹=𝑇₂𝐽𝑃^∗₂,𝐺=(𝑋₁Λ𝑋^†₁)^∗𝐹𝑊₂+𝑅₂𝑌, (2.7) with 𝐽∈𝐶^{(𝑘−𝑔)×(𝑘−𝑡)}, 𝑌∈𝐶^{(𝑘−𝑙)×(𝑘−𝑠)} being arbitrary matrices, and 𝑈 is the same as in (2.1).

Proof. By Lemma 2.1, we know that (𝐾,𝑀) is a solution to Problem 1 if and only if there exist matrices 𝑁,𝐹∈𝐶^𝑘×𝑘 such that 𝑁𝐾=𝑈0𝑁_∗0𝑈^∗𝐹,𝑀=𝑈0𝐹_∗0𝑈^∗,𝑈𝑁0𝑁_∗0𝑈^∗𝐹𝑋=𝑈0𝐹_∗0𝑈^∗𝑋Λ. (2.8)
Using (2.3), the above equation is equivalent to the following two equations: 𝑁𝑋₂=𝐹𝑋₂𝑁Λ,_∗𝑋₁=𝐹^∗𝑋₁Λ,i.e.,𝑋^∗₁𝑁=(𝑋₁Λ)^∗𝐹. (2.9)(2.10)
By the singular value decomposition of 𝑋₂, then the relation (2.9) becomes 0=𝐹𝑋₂Λ𝑉₂,𝑁𝑊₁Σ=𝐹𝑋₂Λ𝑉₁. (2.11)(2.12)
Clearly, (2.11) with respect to unknown matrix 𝐹 is always solvable. By Lemma 2.2 and (2.5), we get 𝐹=𝐿𝑃^∗₂, (2.13) where 𝐿∈𝐶^{𝑘×(𝑘−𝑡)} is an arbitrary matrix. Substituting 𝐹=𝐿𝑃^∗₂ into (2.12), we get 𝑁𝑊₁=(𝐿𝑃^∗₂)𝑋₂Λ𝑉₁Σ⁻¹. (2.14) Since 𝑊₁ 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 𝑁=(𝐿𝑃^∗₂𝑋₂Λ𝑉₁Σ⁻¹)𝑊^∗₁+𝐺𝑊^∗₂=𝐿𝑃^∗₂𝑋₂Λ𝑋^†₂+𝐺𝑊^∗₂, (2.15) where 𝐺∈𝐶^{𝑘×(𝑘−𝑠)} is an arbitrary matrix.
Substituting 𝐹=𝐿𝑃^∗₂ and (2.15) into (2.10), we get 𝑋^∗₁(𝐿𝑃^∗₂𝑋₂Λ𝑋^†₂+𝐺𝑊^∗₂)=(𝑋₁Λ)^∗𝐿𝑃^∗₂. (2.16)
By the singular value decomposition of 𝑋₁, then the relation (2.16) becomes 0=𝑆^∗₂(𝑋₁Λ)^∗𝐿𝑃^∗₂,𝐷𝑅^∗₁(𝐿𝑃^∗₂𝑋₂Λ𝑋^†₂+𝐺𝑊^∗₂)=𝑆^∗₁(𝑋₁Λ)^∗𝐿𝑃^∗₂. (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 𝐿=𝐽₁−(𝑋₁Λ𝑆₂)(𝑋₁Λ𝑆₂)^†𝐽₁𝑃^∗₂𝑃₂=𝐽₁−(𝑋₁Λ𝑆₂)(𝑋₁Λ𝑆₂)^†𝐽₁=𝑇₂𝐽, (2.19) where 𝐽∈𝐶^{(𝑘−𝑔)×(𝑘−𝑡)} is arbitrary. Substituting 𝐿=𝑇₂𝐽 into (2.18), we get 𝐷𝑅^∗₁𝐺𝑊^∗₂=(𝑋₁Λ𝑆₁)^∗𝑇₂𝐽𝑃^∗₂−𝐷𝑅^∗₁𝑇₂𝐽𝑃^∗₂𝑋₂Λ𝑋^†₂. (2.20) Then, we have 𝑅^∗₁𝐺𝑊^∗₂=𝐷⁻¹(𝑋₁Λ𝑆₁)^∗𝑇₂𝐽𝑃^∗₂−𝑅^∗₁𝑇₂𝐽𝑃^∗₂𝑋₂Λ𝑋^†₂. (2.21) 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 𝐺𝑊^∗₂=(𝑋₁Λ𝑋^†₁)^∗𝑇₂𝐽𝑃^∗₂−𝑅₁𝑅^∗₁𝑇₂𝐽𝑃^∗₂𝑋₂Λ𝑋^†₂+(𝐼−𝑅₁𝑅^∗₁)𝑌₁, (2.22) where 𝑌₁∈𝐶^𝑘×𝑘 is arbitrary. Then, we get 𝐺=(𝑋₁Λ𝑋^†₁)^∗𝑇₂𝐽𝑃^∗₂𝑊₂−𝑅₁𝑅^∗₁𝑇₂𝐽𝑃^∗₂𝑋₂Λ𝑋^†₂𝑊₂+(𝐼−𝑅₁𝑅^∗₁)𝑌₁𝑊₂,=(𝑋₁Λ𝑋^†₁)^∗𝑇₂𝐽𝑃^∗₂𝑊₂+𝑅₂𝑌, (2.23) where 𝑌∈𝐶^{(𝑘−𝑙)×(𝑘−𝑠)} is arbitrary.
Finally, we have 𝐹=𝑇₂𝐽𝑃^∗₂,𝑁=𝐹𝑋₂Λ𝑋^†₂+𝐺𝑊^∗₂, (2.24) where 𝐺=(𝑋₁Λ𝑋^†₁)^∗𝐹𝑊₂+𝑅₂𝑌. The proof is completed.

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⁻¹𝐾. 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 be𝑈Γ0𝑋=𝑉^𝑇,𝑈∈𝑂𝑅^𝑛×𝑛,𝑉∈𝑂𝑅^𝑚×𝑚,Γ=diag(𝛾₁,…,𝛾_𝑚)>0, (2.26) where 𝑂𝑅^𝑛×𝑛 denotes the set of all orthogonal matrices. The solution of (2.25) can be expressed as 𝑈Γ𝑉𝐴=^𝑇Λ𝑉Γ⁻¹𝑍₁₂0𝑍₂₂𝑈^𝑇, (2.27) where 𝑍₁₂∈𝑅^{𝑚×(𝑛−𝑚)} is an arbitrary matrix and 𝑍₂₂∈𝑅^{(𝑛−𝑚)×(𝑛−𝑚)} is an arbitrary diagonalizable matrix (see [21, Theorem 3.1]).

Let Λ=diag(𝜆₁𝐼^𝑘1,…,𝜆_𝑞𝐼^𝑘𝑞) with 𝜆₁<𝜆₂<⋯<𝜆_𝑞. Choose 𝑍₂₂=𝐺Λ₂𝐺⁻¹, where 𝐺∈𝑅^{(𝑛−𝑚)×(𝑛−𝑚)} is an arbitrary nonsingular matrix and Λ₂=diag(𝜆_𝑞+1𝐼^𝑘𝑞+1,…,𝜆_𝑝𝐼^𝑘𝑝) 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(𝜆₁𝐼^𝑘1,…,𝜆_𝑞𝐼^𝑘𝑞)∈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 𝑀=𝑈𝐹^𝑇𝐹𝑈^𝑇,𝐾=𝑈𝐹^𝑇Δ𝐹𝑈^𝑇, (2.28) where Δ=diag(Λ,Λ₂), 𝐹=(^𝐹11𝐹12_0𝐹22), 𝐹₁₁=diag(𝐿₁,…,𝐿_𝑞)𝑉Γ⁻¹∈𝑅^𝑚×𝑚, and 𝐹₂₂=diag(𝐿_𝑞+1,…,𝐿_𝑝)𝐺⁻¹∈𝑅^{(𝑛−𝑚)×(𝑛−𝑚)}, where 𝐿_𝑖∈𝑅^{𝑘𝑖×𝑘𝑖} is an arbitrary nonsingular matrix (𝑖=1,2,…,𝑝). The matrix 𝐹₁₂ satisfies the equation Λ𝐹₁₂𝐺−𝐹₁₂𝐺Λ₂=𝐹₁₁𝑍₁₂𝐺.

3. Solution of Problem 2

Lemma 3.1 (see [22]). Given that 𝐴∈𝐶^𝑚×𝑛,𝐵∈𝐶^𝑝×𝑞,𝐶∈𝐶^𝑙×𝑛,𝐷∈𝐶^𝑝×𝑡,𝐸∈𝐶^𝑚×𝑞, and 𝐻∈𝐶^𝑙×𝑡, let 𝑆_𝑎={𝑍∣𝑍∈𝐶^𝑛×𝑝,‖[𝐴𝑍𝐵−𝐸,𝐶𝑍𝐷−𝐻]‖²𝑆=min},_𝑏={𝑍∣𝑍∈𝐶^𝑛×𝑝,𝐴^∗𝐴𝑍𝐵𝐵^∗+𝐶^∗𝐶𝑍𝐷𝐷^∗=𝐴^∗𝐸𝐵^∗+𝐶^∗𝐻𝐷^∗}. (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 𝑈^∗𝑀_𝑎𝐶𝑈=¹𝐶₂𝐶^∗₂𝐶₃,𝑈^∗𝐾_𝑎𝐾𝑈=¹𝐾₂𝐾^∗₂𝐾₃. (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 0𝐹𝐹𝑀=𝑈_∗0𝑈^∗,0𝐾=𝑈𝐹𝑋₂Λ𝑋^†₂+𝐾₂𝑊₂𝑊^∗₂(𝐹𝑋₂Λ𝑋^†₂+𝐾₂𝑊₂𝑊^∗₂)^∗0𝑈^∗, (3.3) where 𝐹=(𝐶₂+𝐾₂(𝑋₂Λ𝑋^†₂)^∗)(𝐼+(𝑋₂Λ𝑋^†₂)(𝑋₂Λ𝑋^†₂)^∗)⁻¹. (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 ‖𝑀_𝑎−𝑀‖²+‖𝐾_𝑎−𝐾‖²𝐶=‖¹𝐶₂𝐶^∗₂𝐶₃−𝐹0𝐹_∗0‖²𝐾+‖¹𝐾₂𝐾^∗₂𝐾₃−0𝐹𝑋₂Λ𝑋^†₂+𝐺𝑊^∗₂(𝐹𝑋₂Λ𝑋^†₂+𝐺𝑊^∗₂)^∗0‖², (3.5) where 𝐺=(𝑋₁Λ𝑋^†₁)^∗𝐹𝑊₂+𝑅₂𝑌. Hence, ‖𝑀_𝑎−𝑀‖²+‖𝐾_𝑎−𝐾‖²=min is equivalent to ‖𝐹−𝐶₂‖²+‖𝐹𝑋₂Λ𝑋^†₂+(𝑋₁Λ𝑋^†₁)^∗𝐹𝑊₂𝑊^∗₂+𝑅₂𝑌𝑊^∗₂−𝐾₂‖²=min. (3.6)
Let 𝑓=‖𝐹−𝐶₂‖²+‖𝐹𝑋₂Λ𝑋^†₂+(𝑋₁Λ𝑋^†₁)^∗𝐹𝑊₂𝑊^∗₂+𝑅₂𝑌𝑊^∗₂−𝐾₂‖². (3.7) Then from the unitary invariant of the Frobenius norm, we have 𝑓=‖𝐹−𝐶₂‖²+‖𝐹𝑋₂Λ𝑋^†₂(𝑊₁,𝑊₂)+(𝑋₁Λ𝑋^†₁)^∗𝐹𝑊₂𝑊^∗₂(𝑊₁,𝑊₂)+𝑅₂𝑌𝑊^∗₂(𝑊₁,𝑊₂)−𝐾₂(𝑊₁,𝑊₂)‖²=‖𝐹−𝐶₂‖²+‖(𝐹𝑋₂Λ𝑋^†₂𝑊₁,0)+(0,(𝑋₁Λ𝑋^†₁)^∗𝐹𝑊₂)+(0,𝑅₂𝑌)−(𝐾₂𝑊₁,𝐾₂𝑊₂)‖²=‖𝐹−𝐶₂‖²+‖𝐹𝑋₂Λ𝑋^†₂𝑊₁−𝐾₂𝑊₁‖²+‖(𝑋₁Λ𝑋^†₁)^∗𝐹𝑊₂+𝑅₂𝑌−𝐾₂𝑊₂‖². (3.8)
Let ℎ=‖(𝑋₁Λ𝑋^†₁)^∗𝐹𝑊₂+𝑅₂𝑌−𝐾₂𝑊₂‖². It is not difficult to see that, when 𝑅₂𝑌=𝐾₂𝑊₂−(𝑋₁Λ𝑋^†₁)^∗𝐹𝑊₂, (3.9) that is, 𝑌=𝑅^∗₂𝐾₂𝑊₂−𝑅^∗₂(𝑋₁Λ𝑋^†₁)^∗𝐹𝑊₂, we have ℎ=0. In other words, we can always find 𝑌 such that ℎ=0. Let 𝑔=‖𝐹−𝐶₂‖²+‖𝐹𝑋₂Λ𝑋^†₂𝑊₁−𝐾₂𝑊₁‖²=‖[𝐹−𝐶₂,𝐹𝑋₂Λ𝑋^†₂𝑊₁−𝐾₂𝑊₁]‖². (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: 𝐹+𝐹(𝑋₂Λ𝑋^†₂𝑊₁)(𝑋₂Λ𝑋^†₂𝑊₁)^∗=𝐶₂+𝐾₂𝑊₁(𝑋₂Λ𝑋^†₂𝑊₁)^∗, (3.11) and its solution is 𝐹=(𝐶₂+𝐾₂(𝑋₂Λ𝑋^†₂)^∗)(𝐼+(𝑋₂Λ𝑋^†₂)(𝑋₂Λ𝑋^†₂)^∗)⁻¹. Again from Lemma 3.1, we have that, when 𝐹=𝐹, 𝑔 attains its minimum, which gives 𝑌=𝑅^∗₂𝐾₂𝑊₂−𝑅^∗₂(𝑋₁Λ𝑋^†₁)^∗𝐹𝑊₂, and 𝐺=(𝑋₁Λ𝑋^†₁)^∗𝐹𝑊₂+𝑅₂𝑌=𝐾₂𝑊₂. 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 3. (1) Input 𝐾_𝑎, 𝑀_𝑎, 𝑋,Λ, and 𝑈.
(2) Compute 𝑋₂ according to (2.3).
(3) Find the singular value decomposition of 𝑋₂ according to (2.4).
(4) Calculate 𝐹 by (3.4).
(5) Compute (𝑀,𝐾) by (3.3).

Example 1. Let 𝑛=6,𝑚=3, and the matrices 𝑀_𝑎,𝐾_𝑎,𝑋, and Λ be given by 𝑀_𝑎=𝐾1.560.660.54−0.39000.660.360.39−0.27000.540.393.1200.54−0.39−0.39−0.2700.720.39−0.27000.540.393.12000−0.39−0.2700.72_𝑎,=23−230036−3300−2−340−2333012−3300−2−3400033012,𝑋=0.0347+0.1507𝑖−0.6975𝑖0.0003+0.0858𝑖0.6715𝑖0.0277+0.0760𝑖−0.0846−0.0101𝑖−0.0009+0.1587𝑖−0.0814+0.0196𝑖0.6967−0.1507+0.0347𝑖0.6975−0.0858+0.0003𝑖−0.6715−0.0760+0.0277𝑖0.0101−0.0846𝑖−0.1587−0.0009𝑖−0.0196−0.0814𝑖0.6967𝑖,Λ=diag(0.3848+0.0126𝑖,2.5545+0.4802𝑖,2.5607). (3.12)(1)(2)(3)
From the Algorithm, we obtain the unique solution of Problem 2 as follows: ,,0𝐹𝐹𝐹=−1.4080+1.1828𝑖1.0322+0.4732𝑖−0.8111−0.0874𝑖0.9537+0.2935𝑖−0.7529−0.0137𝑖−0.6596−0.3106𝑖−0.6624+0.1982𝑖−0.3566−0.0051𝑖−1.0958+1.0040𝑖𝑁=−4.3706+2.1344𝑖1.6264−0.3128𝑖−2.2882−0.3290𝑖2.4251+1.2137𝑖−0.5229+0.0005𝑖−1.4620−0.7688𝑖−1.6669+0.1663𝑖0.6991−0.6057𝑖−2.6437+2.5190𝑖𝑀=𝑈_∗0𝑈^∗,0𝑁𝐾=𝑈𝑈𝑁0^∗, (3.13) where 𝑈=(1/√2)(^𝐼3𝐼3_{−𝑖𝐼3𝑖𝐼3}). It is easy to calculate ‖𝐾𝑋−𝑀𝑋Λ‖=2.1121𝑒−015, and ‖[𝑀−𝑀_𝑎,𝐾−𝐾_𝑎]‖=19.7467.

Acknowledgments

This paper was granted financial support from National Natural Science Foundation (10901056) and Shanghai Natural Science Foundation (09ZR1408700), NSFC grant (10971070). The authors would like to thank the referees for their valuable comments and suggestions.