Toeplitz band matrices with exponentially growing condition numbers

Böttcher, A. ; Grudsky, S.

The electronic journal of linear algebra, Tome 5 (1999), pp. 104-125.

Résumé

Summary: The paper deals with the spectral condition numbers ^(T n (b)) of sequences of Toeplitz matrices T n (b) = (b j$\Gamma k$ ) n j;k=1 as n goes to infinity. The function $b(e i` ) = P$ k b k e ik` is referred to as the symbol of the sequence fT n (b)g. It is well known that ^(T n (b)) may increase exponentially if the symbol b has very strong zeros on the unit circle T = fz 2 C : jzj = 1g, for example, if b vanishes on some subarc of T. If b is a trigonometric polynomial, in which case the matrices T n (b) are band matrices, then b cannot have strong zeros unless it vanishes identically. It is shown that the condition numbers ^(T n (b)) may nevertheless grow exponentially or even faster to infinity. In particular, it is proved that this always happens if b is a trigonometric polynomial which has no zeros on T but nonzero winding number about the origin.

Détail

Classification : 15A12, 15A60, 47B35, 65F35
Keywords: Toeplitz matrix, band matrix, matrix norm, condition number AMS subject