On a condensed form for normal matrices under finite sequences of elementary unitary similarities Original Research Article

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 (or, more precisely, a set of such forms) for general normal matrices, where the number of nonzero elements does not exceed O(n3/2), 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.