Research Article Inverse Eigenvalue Problem of Unitary Hessenberg Matrices

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1 Hindawi Publishing Corporation Discrete Dynamics in Nature and Society Volume 009, Article ID 65069, pages doi:0.55/009/65069 Research Article Inverse Eigenvalue Problem of Unitary Hessenberg Matrices Chunhong Wu and Linzhang Lu, School of Mathematical Sciences, Guizhou Normal University, Xiamen 36005, China School of Mathematics and Computer Science, Xiamen University, Guiyang 55000, China Correspondence should be addressed to Linzhang Lu, Received 9 June 009; Revised 8 August 009; Accepted 3 August 009 Recommended by Binggen Zhang Let H C n n be an n n unitary upper Hessenberg matrix whose subdiagonal elements are all positive, let H be the th leading principal submatrix of H,andlet H be a modified submatrix of H. It is shown that when the minimal and maximal eigenvalues of H,,...,n are nown, H can be constructed uniquely and efficiently. Theoretic analysis, numerical algorithm, and a small example are given. Copyright q 009 C. Wu and L. Lu. 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.. Introduction Direct matrix eigenvalue problems are concerned with deriving and analyzing the spectral information and, hence, predicting the dynamical behavior of a system from a priori nown physical parameters such as mass, length, elasticity, inductance, and capacitance. Inverse eigenvalue problems IEPs, in contrast, are concerned with the determination, identification, or construction of the parameters of a system according to its observed or expected behavior. The inverse eigenvalue problems arise in a remarable variety of applications, such as mathematics physics, control theory, vibration project, structure design, system parameter identification, and the revise of mathematics models 8. Recent years, inverse eigenvalue problem of matrices has become an active topic of computational mathematics for needs of project and technology, and it has resolved a great deal of concrete problem. Especially, the inverse eigenvalue problems have many applications in engineering design, for example, they arise in aviation, civil structure, nucleus engineering, bridge design, shipping construction, and so on. Pole assignment problem have been of major interest in system identification and control theory, we can use optimization techniques to get a solution which is least sensitive to perturbation of problem data. Byrnes 9, Kautsy et al. 0, and

2 Discrete Dynamics in Nature and Society Chu and Li gave an excellent recount of activities in this area. Joseph 7 presented a method for the design of a structure with specified low-order natural frequencies, and the method can further be used to generate initial feasible designs for optimum design problems with frequency constraints. By measuring the changes in the natural frequencies, the IEP idea can be employed to detect the size and location of a blocage in a duct or a crac in a beam, see 5 for additional references. Stare and Inman 6 discussed the applications of IEPs to model updating problems and fault detection problems for machine and structure diagnostics. Applications to other types engineering problems can be found in the boos 4, 7 and articles 8 3. Throughout this paper we use I j to denote the j j identity matrix, e j to denote the jth column of the identity matrix, Λ H to denote the spectrums of a square matrix H, x to denote the complex conjugate of x, andh n to denote the set of unitary upper Hessenberg matrices of order n with positive subdiagonal elements. It is nown 4 that any H H n can be written uniquely as the products H G γ Gn γn G n γn,. where I γ G γ σ σ γ,,,...,n,. I n G n γn diag In, γ n..3 In. and., the parameters γ C,,...,n are called reflection coefficients or Schur parameters in signal processing, σ R,,...,n are said to be complementary parameters and satisfy γ σ, σ > 0,,...,n, and γ n. We refer to. as Schur parametric form of H 5, it plays a fundamental role in the development of efficient algorithms for solving eigenproblems for unitary Hessenberg matrices. However,. is called the complex Givens matrices. H in. is of the explicit form γ σ γ σ σ γ 3 σ σ n γ n σ γ γ γ σ γ 3 γ σ σ n γ n H σ γ γ 3 γ σ 3 σ n γ n, σ n γ n γ n and is uniquely determined by γ,γ,...,γ n. We denote this n n unitary Hessenberg matrix by H γ,γ,...,γ n, each H H n is therefore determined by the n real parameters. Let H be the th leading principal submatrix of H. The matrix H is not unitary for <nand its eigenvalues are inside the unit circle. However, H will become unitary if γ is replaced

3 Discrete Dynamics in Nature and Society 3 by ρ which is any number on the unit circle 4. We introduce the following sequence of modified unitary submatrices: γ σ γ σ σ ρ σ γ γ γ σ σ ρ H ,,,...,n..5 σ γ ρ Because all ρ are of modulus one, the modified submatrices H are unitary and its eigenvalues lie on the unit circle, H H γ,...,γ,ρ. Assume that is not an eigenvalue of H, then j Λ H can be described as j exp iθ j, j,,...,..6 If we number the roots of H starting from π moving counterclocwise along the unit circle, that is, π <θ θ θ π,.7 then we also call exp iθ, λ exp iθ are, respectively, the minimal and maximal eigenvalues of H. Hessenberg matrices arise naturally in several signal processing applications including the frequency estimation procedure and harmonic retrieval problem for radar or sonar navigation 6, 7. Two inds of inverse eigenvalue problems for unitary Hessenberg matrices have been considered up to now. Ammar et al. 8 discussed H H γ,γ,...,γ n H n is uniquely determined by its eigenvalues and the eigenvalues of Ĥ, where Ĥ H αγ,αγ,...,αγ n I α e e T H γ,γ,...,γ n,thatis,ĥ a multiplicative ranone perturbation of H, and the methods are described in 8, 9. Ammar and He in 4 considered that H H n can also be determined by its eigenvalues and the eigenvalues of a modified n n leading principal submatrix of H. In this paper, we consider the following inverse eigenvalue problem. Problem. For n given real numbers θ,θ π, π,,...,n, find unitary Hessenberg matrices H H n, such that exp iθ, λ exp iθ are, respectively, the minimal and the maximal eigenvalues of H for all,,...,n. This paper is organized as follows. In Section, we discussed the properties of unitary Hessenberg matrix. Then the necessary and sufficient conditions for solvability of Problem are derived in Section 3. Section 4 gives the algorithm and numerical example for the problem.

4 4 Discrete Dynamics in Nature and Society. The Properties of Unitary Hessenberg Matrix We denote the characteristic polynomials of H by ϕ,thatis,ϕ λ det λi H. We can appropriately choose ρ such that ϕ λ satisfy the three-term recurrence relations 30, 3, the following lemma give a special method to define ρ. Lemma. see 3. Let H H γ,...,γ n H n, assume is not an eigenvalue of H, define ρ n γ n, ρ γ ρ γ ρ, n,n,...,.. Let H H,...,n be the modified unitary submatrices defined by.5. If one number the eigenvalues of H starting from moving counterclocwise along the unit circle, then the eigenvalues of H interlace those of H in the following sense: the jth eigenvalue of H lies on the arc between the jth and the j st eigenvalue of H. If ρ are defined by., we get the following lemma. Lemma. see 3. The characteristic polynomials ϕ λ det λi H of H defined by.5 satisfy the following three-term recurrence relations: ϕ 0 λ, ϕ λ λ ρ,. ϕ λ λ ρ ρ ϕ λ α λϕ λ,, 3,...,n, where α γ ρ γ ρ ρ γ, γ0..3 Lemma.3. If ρ defined by., α defined by.3, then γ 0, γ α γ ρ, for,,...,n, ρ γ γ n ρ n..4 Proof. By.,weget ρ γ ρ γ ρ,.5 then ρ ρ γ γ ρ..6

5 Discrete Dynamics in Nature and Society 5 Substituting the above formula into.3, weobtain α γ ρ γ γ ρ..7 Because ρ / γ, we have γ α γ ρ ρ γ,,,...,n..8 Lemma.4. Let x C with x and ϕ λ be the characteristic polynomials of H,then ϕ x x ρ ϕ x,,,...,n..9 Proof. It is easy to verify that ϕ x x x x x x x x x λ x x x det H ϕ x.0 ρ ϕ x ρ ϕ x. 3. The Solution of Problem We now consider the solvability conditions of Problem and give the following theorem. Theorem 3.. For n given real number θ,θ π, π,,...,n, there is a unique H γ,γ,...,γ n H n such that exp θ, λ exp are, respectively, the minimal and the maximal eigenvalues of H,,...,n, if and only if π <θ n <θ n < <θ <θ <θ < <θ n n <θ n n π. 3. Proof. Sufficiency. Noticethat π <θ n <θ n < <θ <θ <θ < <θ n n <θ n n π. 3.

6 6 Discrete Dynamics in Nature and Society By Lemma. we have that, if, λ are the eigenvalues of H, they must be the minimal and the maximal eigenvalues of H, respectively. So Problem having a solution is equivalent to that the following equations: ϕ ϕ 0, 0, 3.3 having solutions α, ρ satisfying ρ for all,,...,n. For j, we get ρ λ exp iθ,so ρ. For j n,bylemma., from. and 3.3, we have ρ ρ ϕ ρ ρ ϕ α ϕ α ϕ 0, Then α ϕ α ϕ ρ ρ ϕ ρ ρ ϕ ϕ ϕ,. 3.5 Let m contradiction. ϕ ϕ λ ϕ ϕ, we now show that m / 0by Assume that m 0. Multiplying the first and second equation of 3.5 by ϕ ϕ, respectively, we get λ ϕ ϕ 0,, 3.6 so we obtain by ϕ / 0andϕ / 0. This is a contradiction with /, therefore, m / 0. By ρ, we get ρ m j / 0. Then 3.5 have the unique solution α ρ ρ λ ϕ ϕ, 3.7 ρ m ϕ ϕ ϕ ϕ ρ m. 3.8 We show ρ by induction. By ρ j,,...,. exp iθ,so ρ. Assume that ρ j, for

7 Discrete Dynamics in Nature and Society 7 By 3.8, / 0, and / 0, we have ρ ϕ λ ρ ϕ ϕ ϕ ρ ρ ρ ϕ ρ ρ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ λ ϕ ϕ ϕ ϕ ϕ ϕ ϕ λ ϕ, ϕ ϕ ϕ 3.9 so ρ. Now we have α,,...,n and ρ,,...,n,bylemma.3, we can get γ,for,,...,n. Then we obtain the n n unitary Hessenberg matrix H H γ,γ,...,γ n. Necessity. Suppose that Problem has a unique solution, that is, exp θ, λ exp are, respectively, the minimal and the maximal eigenvalues of H,,...,n, using Lemma.3,weget π <θ n <θ n < <θ <θ <θ < <θ n n <θ n n π. 3.0 Remar 3.. Assume that η 0 is not the eigenvalue of H, we define ρ n γ n, ρ γ η 0 ρ η 0 γ ρ, n,n,...,. 3. Then Lemmas. and. still hold true. 4. Algorithm and Example Based on the above analysis, it is natural that we should propose the following algorithm for solving Problem. Algorithm 4.. Input π <θ n Output H n ; <θ n < <θ <θ <θ Set ρ exp iθ ; Compute α and ρ by 3.7 and 3.8 for, 3,...,n; < <θ n n π;

8 8 Discrete Dynamics in Nature and Society 3 Set γ 0 ; 4 Compute γ by.4 for,,...,n ; 5 Set γ n ρ n. We present an example to illustrate this algorithm. Example 4.. Let n 5, given θ π/6; θ π/8, θ π/4; θ 3 π/4, θ 3 θ 4 π/3, θ 4 4 π/; θ 5 π/, θ 5 5 π/3. By j λ i; exp iθ j λ i, λ i;, weget 3 π/3; λ i, λ i; λ i, λ i; λ i, λ i. 4. Using Algorithm 4., we obtain {ρ i } 5 i, {α i} 4 i, {γ i} 5 i Hessenberg matrix is given as follows: listed in Table. The unitary H i i i i i i i i i i i i i i i 4. We recompute the spectrum of H,,...,n, andget Λ H i, Λ H i, i, Λ H i, i, i, Λ H i, i, i, i, Λ H i, i, i, i, i. 4.3

9 Discrete Dynamics in Nature and Society 9 Table : {ρ i } 5 i, {α i} 4 i, {γ i} 5 i. i ρ i α i γ i i i i i i i i i i i i i i i Figure : The eigenvalues of H. These obtained data show that Algorithm 4. is quite efficient, Figure illustrates the eigenvalues of H,,...,5. Acnowledgment This wor is supported by National Natural Science Foundation of China no References L. R. Fletcher, An inverse eigenvalue problem from control theory, in Numerical Treatment of Inverse Problems in Differential and Integral Equations Heidelberg, 98, P. Deuflhard and E. Hairer, Eds., vol. of Progress in Scientific Computing, pp. 6 70, Birhäuser, Boston, Mass, USA, 983. S. Zhou and H. Dai, The Algebraic Inverse Eigenvalue Problems, Henan Science and Technology Press, Zhenzhou, China, W. M. Wonham, Linear Multivariable Control: A Geometric Approach, vol. 0 of Applications of Mathematics, Springer, New Yor, NY, USA, nd edition, G. M. L. Gladwell, Inverse Problems in Vibration, vol. 9 of Monographs and Textboos on Mechanics of Solids and Fluids: Mechanics. Dynamical Systems, Martinus Nijhoff, Dordrecht, The Netherlands, G. M. L. Gladwell, The inverse problem for the vibrating beam, Proceedings of the Royal Society, vol. 393, no. 805, pp , V. Barcilon, Sufficient conditions for the solution of the inverse problem for a vibrating beam, Inverse Problems, vol. 3, no., pp. 8 93, 987.

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