A survey on inverse and generalized inverse eigenvalue problems for Jacobi matrices

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Abstract

In this paper, we give a survey of well-known works of constructing an specific tridiagonal matrix using prescribed data. Then we focus on two well-known methods namely, m-functions and Lanczos algorithm. We describe the first method for a generalized inverse eigenvalue problem and the second one for a classic inverse eigenvalue problem. We show that these two methods are linked together. In both method we prove that the solution is unique.

Introduction

Direct problems are concerned with predicting the behavior of a given system. Inverse problems are concerned with the determination (or identification)or construction of a system from a knowledge of its behavior. Direct matrix eigenvalue problems are usually well-posed, i.e., any square matrix has a unique set of eigenvalues. Thus, in direct problems main interests are the sensitivity, the stability, convergence and methods of derivation of the eigenvalues from the given matrix. However, inverse eigenvalues are often ill-posed, i.e., given some spectral data there may be non, one, or more than one matrix with the prescribed spectral data. We take advantage of the special structure of tridiagonal matrices. Such matrices satisfy three term recurrence relations which eases the computation of the desired matrix from the given spectral data. In this paper, we present old and new results on inverse and generalized inverse eigenvalue problems for tridiagonal Jacobi matrices. This is a discrete case of Sturm–Liouville problem. In continuous case more than one frequency spectrum is required for reconstruction of a prescribed system, i.e., the coefficient functions in the governing differential equation. We have a similar property for discrete case. For more details see [1], [2], [3].

Section snippets

A survey on well-known results

We start this section with definition of Jacobi matrix. A Jacobi matrix is a tridiagonal matrix of the formA=a1c10···c1a2c20··0c2a3c30············cN-1····cN-1aN,where ci > 0, for i = 1, 2, …, N. Earlier comprehensive results were published by Hald [11] in 1976. He proved that a real Jacobi matrix can be constructed by its eigenvalues and the largest leading principal submatrix. Indeed, he proved the following computation Theorem:

Theorem 1

Let {λi}1N and {μj}1N-1 be two sets of real numbers which satisfy the

Method of m-functions

This method inherited from continuous Sturm-Liouville theory the so-called Weyl m-functions. In 1999, Gesztesy and Simon [5] used these functions to prove uniqueness and to determine the solution of inverse problem corresponding to one dimensional Schródinger operator H=-dh2dx2+q. In 2000, Gerald developed this idea to a Jacobi operator with matrix representation of the form (10). For more detail see the comprehensive monograph by Gerald [12]. Now we use this idea to tackle a finite dimensional

Solving GIEP via m-functions

In this section, we establish a reconstruction procedure to find the unique solution for a generalized inverse eigenvalue problem (GIEP). To do this end we need to find the asymptotic form of the m-functions.

Theorem 6

Function m(z) has the following asymptotic formula aszgoes to infinity:m(z)=-1b1z-a1(b1z)2-(b1b2)c12+a12(b1z)3-O(z-4).

Proof

By Lemma 2 we havem(z,1)=-ψ2(z)c1ψ1(z)=-det(zB[3,N]-A[3,N])det(zB[2,N]-A[2,N]).Therefore, m(z,1)=-1b2z+O(z-2). By recurrence relation (13) relating two successive

Construction of the solution using m-functions

We can now explain the procedure for recovering the matrix A from given spectral function τ(λ), or equivalently m(z)=Rdτ(λ)(λ-b1z), and the diagonal matrix B as follows:

  • (i)

    use the formula given by (20) to recover a1 and c12.

  • (ii)

    Using (13) we obtain,m(z,1)=c1-2a1-b1z-1m(z)to find m(z, 1), which is m-function for the pair (A[2,N], B[2,N]).

  • (iii)

    Go to the step (i) to find a2 and c22, and then (21) to find m(z, 2), …, etc.

In order to illustrate the construction algorithm, we give an example for the case N = 3.

Example 1

Lanczos algorithm

This method is a constructive and numerically well conditioned algorithm based on a given set of spectrum and corresponding normalized eigenvectors. Indeed, we have the following theorem:

Theorem 8

Let λ1 < λ2 <  < λn be a given set of real numbers and let {u(i)}1n be a sequence of normalized eigenvectors such thati=1nu1(i)=1.Then there is a unique Jacobi matrix J such that σ(J)={λi}1n and {u(i)}1n are the corresponding eigenvectors.

This algorithm not only constructs the unique Jacobi matrix J, but also

Concluding remarks

In Lanczos algorithm instead of using one sequence of spectrum and the first components of the corresponding normalized eigenvectors it suffices to consider a pair of the spectrum, σ(A, B) and σ(A[2,N], B[2,N]), and use Theorem 7 to compute the first components of the eigenvectors used in Lanczos algorithm.

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