The Characteristic Roots of Certain Real Symmetric Matrices

The main purpose of this thesis is to collect and coordinate some known results in the study of characteristic roots of certain real symmetric matrices. Specifically, most of the matrices considered have their elements a_ij = 0 except for I i - J I < 1 and are known as Jacobi matrices. The thesis fills in the details of two papers by D. E. Rutherford [12; 13]¹ , with some translation of his results for determinants to the problem of finding characteristic roots of matrices. Some of the results obtained are briefly discussed relative to certain theorems on bounds for characteristic roots.

The first part of the paper is an attempt to show some of the physical sources of the problem and indicate some applications of the results. The aim here is to exhibit matrices of the type considered in the problem and little effort is made to give a detailed theory of the physical laws involved.

A background for the study is the purpose of the next section. A rather general treatment of theorems concerning characteristic roots is given, especially with respect to real symmetric matrices. More recent discoveries concerning bounds of characteristic roots are given. Proofs are either sketched briefly or omitted entirely since most of the theorems are well known.

The main problem is introduced by solving one of the physical problems discussed in the section on applications. The results obtained are used as a basis for a more generalized treatment With the aim of finding the characteristic roots of several classes of n by n matrices.

The matrices to be considered are at best somewhat restricted in scope. This seems to limit the usefulness of the results. However, it is feasible from a practical viewpoint to reduce any real symmetric matrix to a Jacobi matrix in such a way as to preserve the characteristic roots. This fact, [6] , gives added value to the results obtained. Aside from the actual solution of certain physical problems, it is hoped that results here obtained might be of some importance for at least two reasons: first, it makes available several classes of matrices of order n for which the proper values can be obtained from tables of trigonometric functions and, secondly, the methods used here might suggest an approach to the solution of the problem for more general real symmetric matrices . Much remains to be done, even in the special case of the Jacobi matrix.