**2. Problem statements**

**1. Introduction**

28 Applied Linear Algebra in Action

the system.

base the design.

oriented examples.

Many physical problems can be rewritten as generalized inverse eigenvalue problems. For example, in mechanical and structural system design, engineers are often faced with the task of designing systems, which have natural frequencies either that must fall outside a specific range or that must be at exactly certain frequencies. These design problems can be considered as inverse eigenvalue problems because the eigenvalues are directly related to the natural frequencies of

Much focus has been applied to the study of discrete inverse eigenvalue problems. This has been made clear by a thorough review of the topic by Chu and Golub [1, 2]. Gladwell [3] takes a more direct route in which he considers specific inverse problems and matrix structures related to mechanical vibrations. Particularly, it appears that most of the literature focuses on system identification. One of the most common techniques in inverse eigenvalue problems is to use/measure the system's spectrum and then constrain the system in some fashion in order to obtain a second spectrum [4–8]. This clearly indicates that a system usually exists and that it can be tested to obtain data required for the inverse problem of mathematically reconstruct‐ ing the system. Although interesting, this approach cannot be used for novel engineering design to construct a system having a specific spectrum without another system on which to

In most cases, the research on inverse eigenvalue problems has focused on the existence, uniqueness and computability of a solution. Other studies are typical variations on those described above, including partially described problems where not all spectral information is known. These types of problems have been considered in the studies of Gladwell and Willms [9] and Ram and Elhay [10] and are of interest for problems requiring only certain frequencies to be specifically determined. Interestingly, Dias de Silva [11] and de Oliveira [12] have shown that an *n* × *n* matrix always exists when a minimum of *n* − 1 prescribed matrix entries and a prescribed characteristic polynomial are given as design information. The results of Dias de

One of the main shortcomings of current inverse eigenvalue theory is the lack of a solution for general matrices having predefined forms but which do not fit within current known solution methods. In this chapter, two recently proposed methods for structured (direct) solutions of inverse eigenvalue problems are presented. The presented methods are not restricted to matrices of a specific type and are thus applicable to general matrices. For the first method, the Cayley–Hamilton theorem is developed for the generalized eigenvalue vibration problem. The Cayley–Hamilton theorem algorithm is shown to be a good design tool for solving inverse generalized eigenvalue problems. Examples of application of the method are given. A second method, referred to as the inverse eigenvalue determinant method, is also introduced. This method provides another direct approach to the reconstruction of the matrices of the gener‐ alized eigenvalue problem, given knowledge of its eigenvalues and various physical param‐ eters. As for the first method, there are no restrictions on the type of matrices allowed for the inverse problem. Examples of application of the method are also given, including application-

Silva and de Oliveira guarantee existence but not uniqueness of the matrix.

In this chapter, we consider the following problems:

PROBLEM A: *Given a specified set of eigenvalues*, *λ*1, …, *λ<sup>n</sup> construct an n*-*order system*, *described by an n* × *n matrix A*, *which has λ*1, …, *λ<sup>n</sup> as its eigenvalues*.

Although this problem has been considered for specific forms of matrices (i.e. Jacobi, band or other matrix forms), a general solution approach does not currently exist. Problem A leads to the generalized inverse eigenvalue problem, also presented here:

PROBLEM B: *Given a specified set of eigenvalues*, *λ*1, …, *λ<sup>n</sup> construct a system with n*-*degrees of freedom*, *described by two n* × *n matrices* (*M and K*), *which has λ*1, …, *λ<sup>n</sup> as its generalized eigenval‐ ues*: det(*K* − *λM*) = 0 *for the given λ*1, …, *λn*.

For an engineer, Problem B relates directly to the design problem stated earlier, where a conservative vibrating system having specified natural frequencies is sought.

PROBLEM C: *Given a specified set of eigenvalues*, *λ*1, …, *λ<sup>m</sup> construct a system with n degrees of freedom*, *where m* < *n*, *described by two n* × *n matrices* (*M and K*), *which has λ*1, …, *λ<sup>m</sup> as its generalized eigenvalues*: det(*K* − *λM*) = 0 *for the given λ*1, …, *λm*.

Problem C, referred to as a partially described system, is one for which only a select few eigenvalues are specified, as opposed to the entire spectrum being given.
