PUB - Publikationen an der Universität Bielefeld

It is generally known that any Hermitian matrix can be reduced to a tridiagonal form by a finite sequence of unitary similarities, namely Householder reflections. Recently A. Bunse-Gerstner and L. Elsner have found a condensed form to which any unitary matrix can be reduced, again by a finite sequence of Householder transformations. This condensed form can be considered as a pentadiagonal or block tridiagonal matrix with some additional zeros inside the band. We describe such a condensed form for, more precisely, a set of such forms) for general normal matrices, where the number of nonzero elements does not exceed O(n(3/2)), n being the order of the normal matrix given. Two approaches to constructing the condensed form are outlined. The first approach is a geometrical Lanczos-type one where we use the so-called generalized Krylov sequences. The second, more constructive approach is an elimination process using Householder reflections. Our condensed form can be thought of as a variable-bandwidth form. An interesting feature of it is that for normal matrices whose spectra lie on algebraic curves of low degree the bandwidth is much smaller. (C) 1997 Elsevier Science Inc., 1997.