The Calculation of the Eigenvectors of Codiagonal Matrices

In the Givens method for calculating the eigenvalues and eigenvectors of a matrix, a collineatory transformations is constructed which reduces the matrix to codiagonal form. Givens (1954) has given a complete analysis of the problem of finding the eigenvalues and has described a very satisfactory practical procedure for evaluating them. No such analysis has been given for the eigenvectors, though Givens in an unpublished paper has described a procedure which, in his experience, has given accurate results. In this note an analysis of the problem is given, which explains why the straightforward use of the recursions often gives vectors which are catastrophically in error. A method of solution is described which has been used extensively for calculating the vectors on DEUCE. Much of what is written applies equally well to the codiagonal matrices produced by the method of Lanczos (1950), but because this method is usually programmed using floating-point arithmetic, there are one or two additional complications. These will be the subject of a later note.