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Toeplitz band matrices with exponentially growing condition numbers. (English) Zbl 0940.15005

Spectral condition numbers \(\kappa(T_n(a))\) of sequences of Toeplitz matrices \(T_n(a)=(a_{j-k})_{j,k=1}^n\) for n going to infinity are discussed. Here, \(a\) is a rational function on the complex unit circle \(\mathbb T =\{z\in\mathbb C:|z |=1 \}\) and \(\{a_k\}_{k\in\mathbb{Z}}\) the sequence of the Fourier coefficients. It is known that condition numbers may increase exponentially if the symbol \(a\) has very strong zeros on the unit circle, for example if \(a\) vanishes on some subarc of \( \mathbb T\).
If a function \(a\) is a trigonometric polynomial, i.e. \(T_n(a)\) are band matrices, then \(a\) cannot have strong zeros on open arc unless it vanishes identically. It is proved that the condition numbers \(\kappa(T_n(a))\) may nevertheless grow at least exponentially. In particular, it is proved that this always happens if \(a\) is a trigonometric polynomial which has no zeros on \( \mathbb T\) but nonzero winding number about the origin. Two examples of \(a\) are discussed.
The presented method can be generalized to Toeplitz matrices with rational symbols \(a\) and to the condition numbers associated with \(l^p\) norms \((1 \leq p \leq 2)\) instead of only \(l^2\) norm.

MSC:

15A12 Conditioning of matrices
65F35 Numerical computation of matrix norms, conditioning, scaling
15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
15B57 Hermitian, skew-Hermitian, and related matrices
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