Introduction

A direct method for finding characteristic values is one in which values, which are exact apart from the inevitable rounding off errors, are found after a certain finite number of steps. Direct methods may thus be contrasted with iterative methods, which proceed by means of a sequence of approximate values. In this chapter we present one direct method by which the characteristic values of a symmetric matrix may be found, and another method for finding the characteristic values of an unsymmetric matrix. Thus the chapter is not, and is not intended to be, a compendium of methods; for this the reader must look elsewhere.

Both of the methods presented depend on the reduction of the matrix to ‘triple-diagonal form’ The technique applicable to symmetric matrices is described in § 9.3, and that applicable to unsymmetric matrices in § 9.8. When a symmetric matrix has been reduced to a symmetric triple diagonal form its characteristic values may be found by using certain convenient properties of what are called ‘Sturm's sequences’. These are described in §§ 9.4 and 9.5. A method for finding the characteristic vectors of the reduced, triple-diagonal matrix, and the corresponding vectors of the original matrix, is described in §§ 9.6 and 9.7.