ELA, Volume 5, pp. 104-125, December 1999, abstract.

Toeplitz band matrices with exponentially growing condition numbers

A. Boettcher and S. Grudsky


The paper deals with the (spectral) condition numbers
of sequences of finite Toeplitz matrices whose dimensions
go to infinity. It is well known that the condition numbers may
increase exponentially if the symbol of the matrices has very
strong zeros on the unit circle, for example, if the symbol vanishes
on some subarc of the unit circle. If the symbol is a trigonometric
polynomial, in which case the corresponding Toeplitz matrices
are band matrices, then the symbol cannot have strong zeros unless
it vanishes identically. It is shown that the condition numbers
may nevertheless grow exponentially or even faster to infinity
in this case. In particular, it is proved that this always happens
if the symbol is a trigonometric polynomial which has no zeros on
the unit circle but nonzero winding number about the origin.
The techniques employed in this paper are also
applicable to Toeplitz matrices generated by rational
symbols and to the condition numbers associated with
certain Banach space norms instead of the Euclidean norm.