Inverse Eigenvalue Problem of Unitary Hessenberg Matrices

Abstract

Let 𝐻∈ℂ^𝑛×𝑛 be an 𝑛×𝑛 unitary upper Hessenberg matrix whose subdiagonal elements are all positive, let 𝐻_𝑘 be the 𝑘th leading principal submatrix of 𝐻, and let ^0𝑥0𝐻^02𝑑𝑐_𝑘 be a modified submatrix of 𝐻_𝑘. It is shown that when the minimal and maximal eigenvalues of ^0𝑥0𝐻^02𝑑𝑐_𝑘 (𝑘=1,2,…,𝑛) are known, 𝐻 can be constructed uniquely and efficiently. Theoretic analysis, numerical algorithm, and a small example are given.

1. 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 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 [9], Kautsky et al. [10], and Chu and Li [11] 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 blockage in a duct or a crack in a beam, see [12–15] for additional references. Starek and Inman [16] 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 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, and ℋ_𝑛 to denote the set of unitary upper Hessenberg matrices of order 𝑛 with positive subdiagonal elements.

It is known [24] that any 𝐻∈ℋ_𝑛 can be written uniquely as the products 𝐻=𝐺₁(𝛾₁)⋯𝐺_𝑛−1(𝛾_𝑛−1)𝐺_𝑛(𝛾_𝑛), (1.1) where 𝐺_𝑘(𝛾_𝑘𝐼)=^𝑘−1−𝛾_𝑘𝜎_𝑘𝜎_𝑘𝛾_𝑘𝐼_{𝑛−𝑘−1}𝐺,𝑘=1,2,…,𝑛−1,_𝑛(𝛾_𝑛)=diag(𝐼_𝑛−1,−𝛾_𝑛). (1.2)(1.3) In (1.1) and (1.2), the parameters 𝛾_𝑘∈ℂ (𝑘=1,2,…,𝑛) are called reflection coefficients or Schur parameters in signal processing, 𝜎_𝑘∈ℝ (𝑘=1,2,…,𝑛−1) are said to be complementary parameters and satisfy |𝛾_𝑘|²+𝜎²_𝑘=1, 𝜎_𝑘>0, 𝑘=1,…,𝑛−1, and |𝛾_𝑛|=1. We refer to (1.1) as Schur parametric form of 𝐻 [25], 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. 𝐻 in (1.1) is of the explicit form 𝐻=−𝛾₁−𝜎₁𝛾₂−𝜎₁𝜎₂𝛾₃⋯−𝜎₁⋯𝜎_𝑛−1𝛾_𝑛𝜎₁−𝛾₁𝛾₂−𝛾₁𝜎₂𝛾₃⋯−𝛾₁𝜎₂⋯𝜎_𝑛−1𝛾_𝑛𝜎₂−𝛾₂𝛾₃⋯−𝛾₂𝜎₃⋯𝜎_𝑛−1𝛾_𝑛𝜎⋱⋱⋮_𝑛−1−𝛾_𝑛−1𝛾_𝑛, (1.4) and is uniquely determined by 𝛾₁,𝛾₂,…,𝛾_𝑛. We denote this 𝑛×𝑛 unitary Hessenberg matrix by 𝐻(𝛾₁,𝛾₂,…,𝛾_𝑛), each 𝐻∈ℋ_𝑛 is therefore determined by the 2𝑛−1 real parameters. Let 𝐻_𝑘 be the 𝑘th leading principal submatrix of 𝐻. The matrix 𝐻_𝑘 is not unitary for 𝑘<𝑛 and its eigenvalues are inside the unit circle. However, 𝐻_𝑘 will become unitary if 𝛾_𝑘 is replaced by 𝜌_𝑘 which is any number on the unit circle [24]. We introduce the following sequence of modified unitary submatrices: 𝐻_𝑘=−𝛾₁−𝜎₁𝛾₂⋯−𝜎₁⋯𝜎_𝑘−1𝜌_𝑘𝜎₁−𝛾₁𝛾₂⋯−𝛾₁𝜎₂⋯𝜎_𝑘−1𝜌_𝑘𝜎⋱⋱⋮_𝑘−1−𝛾_𝑘−1𝜌_𝑘,𝑘=1,2,…,𝑛. (1.5) Because all 𝜌_𝑘 are of modulus one, the modified submatrices 𝐻_𝑘 are unitary and its eigenvalues lie on the unit circle, 𝐻_𝑘=𝐻(𝛾₁,…,𝛾_𝑘−1,𝜌_𝑘). Assume that −1 is not an eigenvalue of 𝐻, then 𝜆_𝑗^(𝑘)∈Λ(𝐻_𝑘) can be described as 𝜆_𝑗^(𝑘)=exp(𝑖𝜃_𝑗^(𝑘)),𝑗=1,2,…,𝑘. (1.6) If we number the roots of 𝐻_𝑘 starting from −𝜋 moving counterclockwise along the unit circle, that is, −𝜋<𝜃₁^(𝑘)≤𝜃₂^(𝑘)≤⋯≤𝜃_𝑘^(𝑘)≤𝜋, (1.7) then we also call 𝜆₁^(𝑘)=exp(𝑖𝜃₁^(𝑘)), 𝜆_𝑘^(𝑘)=exp(𝑖𝜃_𝑘^(𝑘)) are, respectively, the minimal and maximal eigenvalues of 𝐻_𝑘.

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. [28] discussed 𝐻=𝐻(𝛾₁,𝛾₂,…,𝛾_𝑛)∈ℋ_𝑛 is uniquely determined by its eigenvalues and the eigenvalues of 𝐻, where 𝐻=𝐻(𝛼𝛾₁,𝛼𝛾₂,…,𝛼𝛾_𝑛)=(𝐼−(1−𝛼)𝑒₁𝑒^𝑇₁)𝐻(𝛾₁,𝛾₂,…,𝛾_𝑛), that is, 𝐻 a multiplicative rank-one perturbation of 𝐻, and the methods are described in [28, 29]. Ammar and He in [24] considered that 𝐻∈ℋ_𝑛 can also be determined by its eigenvalues and the eigenvalues of a modified (𝑛−1)×(𝑛−1) leading principal submatrix of 𝐻.

In this paper, we consider the following inverse eigenvalue problem.

Problem 1. For 2𝑛−1 given real numbers 𝜃₁^(𝑘),𝜃_𝑘^(𝑘)∈(−𝜋,𝜋] (𝑘=1,2,…,𝑛), find unitary Hessenberg matrices 𝐻∈ℋ_𝑛, such that 𝜆₁^(𝑘)=exp(𝑖𝜃₁^(𝑘)), 𝜆_𝑘^(𝑘)=exp(𝑖𝜃_𝑘^(𝑘)) are, respectively, the minimal and the maximal eigenvalues of 𝐻_𝑘 for all 𝑘=1,2,…,𝑛.

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 𝐻_𝑘 by 𝜑_𝑘, that is, 𝜑_𝑘(𝜆)=det(𝜆𝐼_𝑘−𝐻_𝑘). We can appropriately choose 𝜌_𝑘 such that 𝜑_𝑘(𝜆) satisfy the three-term recurrence relations [30, 31], the following lemma give a special method to define 𝜌_𝑘.

Lemma 2.1 (see [32]). Let 𝐻=𝐻(𝛾₁,…,𝛾_𝑛)∈ℋ_𝑛, assume −1 is not an eigenvalue of 𝐻, define 𝜌_𝑛=𝛾_𝑛,𝜌_𝑘=𝛾_𝑘−𝜌_𝑘+11−𝛾_𝑘𝜌_𝑘+1,𝑘=𝑛−1,𝑛−2,…,1. (2.1) Let 𝐻_𝑘∈ℋ_𝑘 (𝑘=1,…,𝑛) be the modified unitary submatrices defined by (1.5). If one number the eigenvalues of 𝐻_𝑘 starting from −1 moving counterclockwise along the unit circle, then the eigenvalues of 𝐻_𝑘 interlace those of 𝐻_𝑘+1 in the following sense: the 𝑗th eigenvalue of 𝐻_𝑘 lies on the arc between the 𝑗th and the 𝑗+1st eigenvalue of 𝐻_𝑘+1.

If 𝜌_𝑘 are defined by (2.1), we get the following lemma.

Lemma 2.2 (see [32]). The characteristic polynomials 𝜑_𝑘(𝜆)=det(𝜆𝐼_𝑘−𝐻_𝑘) of 𝐻_𝑘 defined by (1.5) satisfy the following three-term recurrence relations: 𝜑₀𝜑(𝜆)=1,₁(𝜆)=𝜆+𝜌₁,𝜑_𝑘(𝜆)=(𝜆+𝜌_𝑘𝜌_𝑘−1)𝜑_𝑘−1(𝜆)−𝛼_𝑘−1𝜆𝜑_𝑘−2(𝜆),𝑘=2,3,…,𝑛, (2.2) where 𝛼_𝑘=𝛾_𝑘−1(𝜌_𝑘−𝛾_𝑘)+𝜌_𝑘+1(𝜌_𝑘−𝛾_𝑘),𝛾₀=1. (2.3)

Lemma 2.3. If 𝜌_𝑘 defined by (2.1), 𝛼_𝑘 defined by (2.3), then 𝛾₀𝛾=1,_𝑘=𝛼_𝑘−𝛾_𝑘−1𝜌_𝑘+1𝜌_𝑘−𝛾_𝑘−1𝛾,for𝑘=1,2,…,𝑛−1,_𝑛=𝜌_𝑛. (2.4)

Proof. By (2.1), we get 𝜌_𝑘(1−𝛾_𝑘𝜌_𝑘+1)=𝛾_𝑘−𝜌_𝑘+1, (2.5) then 𝜌_𝑘+1(𝜌_𝑘−𝛾_𝑘)=𝛾_𝑘𝜌_𝑘−1. (2.6) Substituting the above formula into (2.3), we obtain 𝛼_𝑘=𝛾_𝑘−1(𝜌_𝑘−𝛾_𝑘)+𝛾_𝑘𝜌_𝑘−1. (2.7) Because 𝜌_𝑘≠𝛾_𝑘−1, we have 𝛾_𝑘=𝛼_𝑘−𝛾_𝑘−1𝜌_𝑘+1𝜌_𝑘−𝛾_𝑘−1,𝑘=1,2,…,𝑛−1. (2.8)

Lemma 2.4. Let 𝑥∈ℂ with |𝑥|=1 and 𝜑_𝑘(𝜆) be the characteristic polynomials of 𝐻_𝑘, then 𝜑_𝑘(𝑥)𝑥^𝑘=𝜌_𝑘𝜑_𝑘(𝑥),𝑘=1,2,…,𝑛. (2.9)

Proof. It is easy to verify that 𝜑_𝑘(𝑥)𝑥^𝑘=(𝑥−𝜆₁^(𝑘))(𝑥−𝜆₂^(𝑘))⋯(𝑥−𝜆_𝑘^(𝑘))𝑥^𝑘=(1−𝑥𝜆₁^(𝑘))(1−𝑥𝜆₁^(𝑘))⋯(1−𝑥𝜆_𝑘^(𝑘))=𝜆₁^(𝑘)𝜆₂^(𝑘)⋯𝜆_𝑘^(𝑘)(𝜆₁^(𝑘)−𝑥)(𝜆₂^(𝑘)−𝑥)⋯(𝜆_𝑘^(𝑘)−𝑥)=(−1)^𝑘det(𝐻_𝑘)𝜑_𝑘(𝑥)=(−1)^𝑘(−1)^𝑘𝜌_𝑘𝜑_𝑘(𝑥)=𝜌_𝑘𝜑_𝑘(𝑥). (2.10)

3. The Solution of Problem 1

We now consider the solvability conditions of Problem 1 and give the following theorem.

Theorem 3.1. For 2𝑛−1 given real number 𝜃₁^(𝑘),𝜃_𝑘^(𝑘)∈(−𝜋,𝜋](𝑘=1,2,…,𝑛), there is a unique 𝐻(𝛾₁,𝛾₂,…,𝛾_𝑛)∈ℋ_𝑛 such that 𝜆₁^(𝑘)=exp(𝜃₁^(𝑘)), 𝜆_𝑘^(𝑘)=exp(𝜆_𝑘^(𝑘)) are, respectively, the minimal and the maximal eigenvalues of 𝐻_𝑘(𝑘=1,2,…,𝑛), if and only if −𝜋<𝜃₁^(𝑛)<𝜃₁^(𝑛−1)<⋯<𝜃₁⁽²⁾<𝜃₁⁽¹⁾<𝜃₂⁽²⁾<⋯<𝜃^(𝑛−1)_𝑛−1<𝜃_𝑛^(𝑛)≤𝜋. (3.1)

Proof. Sufficiency. Notice that −𝜋<𝜃₁^(𝑛)<𝜃₁^(𝑛−1)<⋯<𝜃₁⁽²⁾<𝜃₁⁽¹⁾<𝜃₂⁽²⁾<⋯<𝜃^(𝑛−1)_𝑛−1<𝜃_𝑛^(𝑛)≤𝜋. (3.2) By Lemma 2.1 we have that, if 𝜆₁^(𝑘), 𝜆_𝑘^(𝑘) are the eigenvalues of 𝐻_𝑘, they must be the minimal and the maximal eigenvalues of 𝐻_𝑘, respectively. So Problem 1 having a solution is equivalent to that the following equations: 𝜑_𝑘(𝜆₁^(𝑘)𝜑)=0,_𝑘(𝜆_𝑘^(𝑘))=0, (3.3) having solutions 𝛼_𝑘−1, 𝜌_𝑘 satisfying |𝜌_𝑘|=1 for all 𝑘=1,2,…,𝑛.
For 𝑗=1, we get 𝜌₁=𝜆₁⁽¹⁾=exp(𝑖𝜃₁⁽¹⁾), so |𝜌₁|=1.
For 2≤𝑗≤𝑛, by Lemma 2.1, from (2.2) and (3.3), we have (𝜆₁^(𝑘)+𝜌_𝑘𝜌_𝑘−1)𝜑_𝑘−1(𝜆₁^(𝑘))−𝛼_𝑘−1𝜆₁^(𝑘)𝜑_𝑘−2(𝜆₁^(𝑘))=0,(𝜆_𝑘^(𝑘)+𝜌_𝑘𝜌_𝑘−1)𝜑_𝑘−1(𝜆_𝑘^(𝑘))−𝛼_𝑘−1𝜆_𝑘^(𝑘)𝜑_𝑘−2(𝜆_𝑘^(𝑘))=0. (3.4) Then 𝛼_𝑘−1𝜆₁^(𝑘)𝜑_𝑘−2(𝜆₁^(𝑘))−𝜌_𝑘𝜌_𝑘−1𝜑_𝑘−1(𝜆₁^(𝑘))=𝜆₁^(𝑘)𝜑_𝑘−1(𝜆₁^(𝑘)𝛼),_𝑘−1𝜆_𝑘^(𝑘)𝜑_𝑘−2(𝜆_𝑘^(𝑘))−𝜌_𝑘𝜌_𝑘−1𝜑_𝑘−1(𝜆_𝑘^(𝑘))=𝜆_𝑘^(𝑘)𝜑_𝑘−1(𝜆_𝑘^(𝑘)). (3.5) Let 𝑚_𝑘≡𝜆₁^(𝑘)𝜑_𝑘−2(𝜆₁^(𝑘))𝜑_𝑘−1(𝜆_𝑘^(𝑘))−𝜆_𝑘^(𝑘)𝜑_𝑘−2(𝜆_𝑘^(𝑘))𝜑_𝑘−1(𝜆₁^(𝑘)), we now show that 𝑚_𝑘≠0 by contradiction.
Assume that 𝑚_𝑘=0. Multiplying the first and second equation of (3.5) by 𝜑_𝑘−1(𝜆_𝑘^(𝑘)), 𝜑_𝑘−1(𝜆₁^(𝑘)), respectively, we get (𝜆₁^(𝑘)−𝜆_𝑘^(𝑘))𝜑_𝑘−1(𝜆₁^(𝑘))𝜑_𝑘−1(𝜆_𝑘^(𝑘))=0, (3.6) so we obtain 𝜆₁^(𝑘)=𝜆_𝑘^(𝑘) by 𝜑_𝑘−1(𝜆₁^(𝑘))≠0 and 𝜑_𝑘−1(𝜆_𝑘^(𝑘))≠0. This is a contradiction with 𝜆₁^(𝑘)≠𝜆_𝑘^(𝑘), therefore, 𝑚_𝑘≠0. By |𝜌_𝑘−1|=1, we get −𝜌_𝑘−1𝑚_𝑗≠0. Then (3.5) have the unique solution 𝛼_𝑘−1=𝜌_𝑘−1(𝜆_𝑘^(𝑘)−𝜆₁^(𝑘))𝜑_𝑘−1(𝜆₁^(𝑘))𝜑_𝑘−1(𝜆_𝑘^(𝑘))−𝜌_𝑘−1𝑚_𝑘𝜌_𝑘,=𝜆₁^(𝑘)𝜆_𝑘^(𝑘)(𝜑_𝑘−2(𝜆₁^(𝑘))𝜑_𝑘−1(𝜆_𝑘^(𝑘))−𝜑_𝑘−2(𝜆_𝑘^(𝑘))𝜑_𝑘−1(𝜆₁^(𝑘)))−𝜌_𝑘−1𝑚_𝑘. (3.7)(3.8) We show |𝜌_𝑘|=1 by induction. By 𝜌₁=exp(𝑖𝜃₁⁽¹⁾), so |𝜌₁|=1. Assume that |𝜌_𝑗|=1, for 𝑗=1,2,…,𝑘−1.
By (3.8), 𝜆₁^(𝑘)≠0, and 𝜆_𝑘^(𝑘)≠0, we have 𝜌_𝑘=𝜆₁^(𝑘)𝜆_𝑘^(𝑘)(𝜑_𝑘−2(𝜆₁^(𝑘))𝜑_𝑘−1(𝜆_𝑘^(𝑘))−𝜑_𝑘−2(𝜆_𝑘^(𝑘))𝜑_𝑘−1(𝜆₁^(𝑘)))(𝜆₁^(𝑘))^𝑘−1(𝜆_𝑘^(𝑘))^𝑘−1−𝜌_𝑘−1(𝜆₁^(𝑘)𝜑_𝑘−2(𝜆₁^(𝑘))𝜑_𝑘−1(𝜆_𝑘^(𝑘))−𝜆_𝑘^(𝑘)𝜑_𝑘−2(𝜆_𝑘^(𝑘))𝜑_𝑘−1(𝜆₁^(𝑘)))(𝜆₁^(𝑘))^𝑘−1(𝜆_𝑘^(𝑘))^𝑘−1=(𝜆₁^(𝑘))^𝑘−2(𝜆_𝑘^(𝑘))^𝑘−2(𝜑_𝑘−2(𝜆₁^(𝑘))𝜑_𝑘−1(𝜆_𝑘^(𝑘))−𝜑_𝑘−2(𝜆_𝑘^(𝑘))𝜑_𝑘−1(𝜆₁^(𝑘)))−𝜌_𝑘−1𝜌_𝑘−2𝜌_𝑘−1(𝜑_𝑘−2(𝜆₁^(𝑘))𝜑_𝑘−1(𝜆_𝑘^(𝑘))−𝜑_𝑘−2(𝜆_𝑘^(𝑘))𝜑_𝑘−1(𝜆₁^(𝑘)=())𝜆₁^(𝑘))^𝑘−2(𝜆_𝑘^(𝑘))^𝑘−2(𝜑_𝑘−2(𝜆₁^(𝑘))𝜑_𝑘−1(𝜆_𝑘^(𝑘))−𝜑_𝑘−2(𝜆_𝑘^(𝑘))𝜑_𝑘−1(𝜆₁^(𝑘)))|𝜌_𝑘−1|²𝜌_𝑘−2(𝜑_𝑘−2(𝜆₁^(𝑘))𝜑_𝑘−1(𝜆_𝑘^(𝑘))−𝜑_𝑘−2(𝜆_𝑘^(𝑘))𝜑_𝑘−1(𝜆₁^(𝑘),)) (3.9) so |𝜌_𝑘|=1.
Now we have 𝛼_𝑘(𝑘=1,2,…,𝑛−1) and 𝜌_𝑘(𝑘=1,2,…,𝑛), by Lemma 2.3, we can get 𝛾_𝑘, for 𝑘=1,2,…,𝑛. Then we obtain the 𝑛×𝑛 unitary Hessenberg matrix 𝐻=𝐻(𝛾₁,𝛾₂,…,𝛾_𝑛).
Necessity. Suppose that Problem 1 has a unique solution, that is, 𝜆₁^(𝑘)=exp(𝜃₁^(𝑘)), 𝜆_𝑘^(𝑘)=exp(𝜆_𝑘^(𝑘)) are, respectively, the minimal and the maximal eigenvalues of 𝐻_𝑘(𝑘=1,2,…,𝑛), using Lemma 2.3, we get −𝜋<𝜃₁^(𝑛)<𝜃₁^(𝑛−1)<⋯<𝜃₁⁽²⁾<𝜃₁⁽¹⁾<𝜃₂⁽²⁾<⋯<𝜃^(𝑛−1)_𝑛−1<𝜃_𝑛^(𝑛)≤𝜋. (3.10)

Remark 3.2. Assume that 𝜂₀ is not the eigenvalue of 𝐻, we define 𝜌_𝑛=𝛾_𝑛,𝜌_𝑘=𝛾_𝑘+𝜂₀𝜌_𝑘+11+𝜂₀𝛾_𝑘𝜌_𝑘+1,𝑘=𝑛−1,𝑛−2,…,1. (3.11) Then Lemmas 2.1 and 2.2 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 1.

Algorithm 4.1. Input −𝜋<𝜃₁^(𝑛)<𝜃₁^(𝑛−1)<⋯<𝜃₁⁽²⁾<𝜃₁⁽¹⁾<𝜃₂⁽²⁾<⋯<𝜃_𝑛^(𝑛)≤𝜋;
Output 𝐻_𝑛; (1)Set 𝜌₁=exp(𝑖𝜃₁⁽¹⁾); (2)Compute 𝛼_𝑘−1 and 𝜌_𝑘 by (3.7) and (3.8) for 𝑘=2,3,…,𝑛; (3)Set 𝛾₀=1; (4)Compute 𝛾_𝑘 by (2.4) for 𝑘=1,2,…,𝑛−1; (5)Set 𝛾_𝑛=𝜌_𝑛.
We present an example to illustrate this algorithm.

Example 4.2. Let 𝑛=5, given 𝜃₁⁽¹⁾=𝜋/6; 𝜃₁⁽²⁾=−𝜋/8, 𝜃₂⁽²⁾=𝜋/4; 𝜃₁⁽³⁾=−𝜋/4, 𝜃₃⁽³⁾=𝜋/3; 𝜃₁⁽⁴⁾=−𝜋/3, 𝜃₄⁽⁴⁾=𝜋/2; 𝜃₁⁽⁵⁾=−𝜋/2, 𝜃₅⁽⁵⁾=2𝜋/3. By 𝜆_𝑗^(𝑘)=exp(𝑖𝜃_𝑗^(𝑘)), we get 𝜆₁⁽¹⁾𝜆=0.8660+0.5000𝑖;₁₍₂₎=0.9239−0.3827𝑖,𝜆₂⁽²⁾𝜆=0.7071+0.7071𝑖;₁₍₃₎=0.7071−0.7071𝑖,𝜆₃⁽³⁾𝜆=0.5000+0.8660𝑖;₁₍₄₎=0.5000−0.8660𝑖,𝜆₄⁽⁴⁾𝜆=0.0000+1.0000𝑖;₁₍₅₎=0.0000−1.0000𝑖,𝜆₅⁽⁵⁾=−0.5000+0.8660𝑖. (4.1) Using Algorithm 4.1, we obtain {𝜌_𝑖}⁵_𝑖=1, {𝛼_𝑖}⁴_𝑖=1, {𝛾_𝑖}⁵_𝑖=1 listed in Table 1. The unitary Hessenberg matrix is given as follows: .𝐻=0.7588+0.4471𝑖−0.3354−0.1185𝑖0.1490+0.0810𝑖0.0045−0.0151𝑖0.1169+0.2346𝑖0.47360.6493−0.1269𝑖−0.3152+0.0109𝑖0.0071+0.0283𝑖−0.4088−0.2656𝑖00.66010.4024+0.0643𝑖−0.0020−0.0377𝑖0.4526+0.4382𝑖000.84010.0068+0.0317𝑖−0.4436−0.3106𝑖0000.99820.0438−0.0407𝑖 (4.2) We recompute the spectrum of 𝐻_𝑘(𝑘=1,2,…,𝑛), and get Λ(𝐻₁)=(0.8660+0.5000𝑖𝐻),Λ(₂)=(0.9239−0.3827𝑖,0.7071+0.7071𝑖𝐻),Λ(₃)=(0.7071−0.7071𝑖,0.9013+0.4332𝑖,0.5000+0.8660𝑖𝐻),Λ(₄)=(0.5000−0.8660𝑖,0.9964−0.0850𝑖,0.7329+0.6803𝑖,0.0000+1.0000𝑖𝐻),Λ(₅)=(0.0000−1.0000𝑖,0.7937−0.6083𝑖,0.9411+0.3381𝑖,0.6262+0.7796𝑖,−0.5000+0.8660𝑖). (4.3) These obtained data show that Algorithm 4.1 is quite efficient, Figure 1 illustrates the eigenvalues of 𝐻_𝑘(𝑘=1,2,…,5).

Table 1 

{𝝆_𝐢}^𝟓_𝐢=𝟏,{𝜶_𝐢}^𝟒_𝐢=𝟏,{𝜸_𝐢}^𝟓_𝐢=𝟏.

Figure 1 

The eigenvalues of 𝐻𝑘.

Acknowledgment

This work is supported by National Natural Science Foundation of China no. 10531080.