Research Article | Open Access
Inverse Eigenvalue Problem of Unitary Hessenberg Matrices
Let be an unitary upper Hessenberg matrix whose subdiagonal elements are all positive, let be the leading principal submatrix of , and let be a modified submatrix of . It is shown that when the minimal and maximal eigenvalues of () are known, can be constructed uniquely and efficiently. Theoretic analysis, numerical algorithm, and a small example are given.
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 known 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 remarkable variety of applications, such as mathematics physics, control theory, vibration project, structure design, system parameter identification, and the revise of mathematics models [1–8]. 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 , Kautsky et al. , and Chu and Li  gave an excellent recount of activities in this area. Joseph  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 blockage in a duct or a crack in a beam, see [12–15] for additional references. Starek and Inman  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 books [4, 17] and articles [18–23].
Throughout this paper we use to denote the identity matrix, to denote the th column of the identity matrix, to denote the spectrums of a square matrix , to denote the complex conjugate of , and to denote the set of unitary upper Hessenberg matrices of order with positive subdiagonal elements.
It is known  that any can be written uniquely as the products where In (1.1) and (1.2), the parameters () are called reflection coefficients or Schur parameters in signal processing, () are said to be complementary parameters and satisfy , , , and . We refer to (1.1) as Schur parametric form of , it plays a fundamental role in the development of efficient algorithms for solving eigenproblems for unitary Hessenberg matrices. However, (1.2) is called the complex Givens matrices. in (1.1) is of the explicit form and is uniquely determined by . We denote this unitary Hessenberg matrix by , each is therefore determined by the real parameters. Let be the th leading principal submatrix of . The matrix is not unitary for and its eigenvalues are inside the unit circle. However, will become unitary if is replaced by which is any number on the unit circle . We introduce the following sequence of modified unitary submatrices: Because all are of modulus one, the modified submatrices are unitary and its eigenvalues lie on the unit circle, . Assume that is not an eigenvalue of , then can be described as If we number the roots of starting from moving counterclockwise along the unit circle, that is, then we also call , are, respectively, the minimal and maximal eigenvalues of .
Hessenberg matrices arise naturally in several signal processing applications including the frequency estimation procedure and harmonic retrieval problem for radar or sonar navigation [26, 27]. Two kinds of inverse eigenvalue problems for unitary Hessenberg matrices have been considered up to now. Ammar et al.  discussed is uniquely determined by its eigenvalues and the eigenvalues of , where , that is, a multiplicative rank-one perturbation of , and the methods are described in [28, 29]. Ammar and He in  considered that can also be determined by its eigenvalues and the eigenvalues of a modified leading principal submatrix of .
In this paper, we consider the following inverse eigenvalue problem.
Problem 1. For given real numbers (), find unitary Hessenberg matrices , such that , are, respectively, the minimal and the maximal eigenvalues of for all
This paper is organized as follows. In Section 2, we discussed the properties of unitary Hessenberg matrix. Then the necessary and sufficient conditions for solvability of Problem 1 are derived in Section 3. Section 4 gives the algorithm and numerical example for the problem.
2. The Properties of Unitary Hessenberg Matrix
We denote the characteristic polynomials of by , that is, . We can appropriately choose such that satisfy the three-term recurrence relations [30, 31], the following lemma give a special method to define .
Lemma 2.1 (see ). Let , assume is not an eigenvalue of , define Let () be the modified unitary submatrices defined by (1.5). If one number the eigenvalues of starting from moving counterclockwise along the unit circle, then the eigenvalues of interlace those of in the following sense: the th eigenvalue of lies on the arc between the th and the st eigenvalue of .
If are defined by (2.1), we get the following lemma.
Lemma 2.4. Let with and be the characteristic polynomials of , then
Proof. It is easy to verify that
3. The Solution of Problem 1
We now consider the solvability conditions of Problem 1 and give the following theorem.
Theorem 3.1. For given real number , there is a unique such that , are, respectively, the minimal and the maximal eigenvalues of , if and only if
Proof. Sufficiency. Notice that
By Lemma 2.1 we have that, if , are the eigenvalues of , they must be the minimal and the maximal eigenvalues of , respectively. So Problem 1 having a solution is equivalent to that the following equations:
having solutions , satisfying for all .
For , we get , so .
For , by Lemma 2.1, from (2.2) and (3.3), we have Then Let , we now show that by contradiction.
Assume that . Multiplying the first and second equation of (3.5) by , , respectively, we get so we obtain by and . This is a contradiction with , therefore, . By , we get . Then (3.5) have the unique solution We show by induction. By , so . Assume that , for .
By (3.8), , and , we have so .
Now we have and , by Lemma 2.3, we can get , for . Then we obtain the unitary Hessenberg matrix .
Necessity. Suppose that Problem 1 has a unique solution, that is, , are, respectively, the minimal and the maximal eigenvalues of , using Lemma 2.3, we get
4. Algorithm and Example
Based on the above analysis, it is natural that we should propose the following algorithm for solving Problem 1.
Example 4.2. Let , given ; , ; , ; , ; , . By , we get Using Algorithm 4.1, we obtain , , listed in Table 1. The unitary Hessenberg matrix is given as follows: We recompute the spectrum of , and get These obtained data show that Algorithm 4.1 is quite efficient, Figure 1 illustrates the eigenvalues of .
This work is supported by National Natural Science Foundation of China no. 10531080.
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Copyright © 2009 Chunhong Wu and Linzhang 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 work is properly cited.