Abstract
The modern version of the Kreiss matrix theorem states that ∥A n∥ ≤esK (for alln≥0) ifA is ans ×s matrix satisfying a resolvent condition with respect to the unit disk with constantK. In this paper we show for any fixedK ≥ π + 1 that the upper boundesK is sharp in the sense that a linear dependence on the dimensions is the best possible. An analogous result is obtained for the continuous version of the Kreiss matrix theorem, which states that ∥exp(tA)∥ ≤esK (for allt≥0) ifA is ans ×s matrix satisfying a resolvent condition with respect to the left half plane with constantK.
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The research has been made possible by a fellowship of the Royal Netherlands Academy of Arts and Sciences (K.N.A.W.) and was carried out at the Department of Mathematics and Computer Science, University of Leiden, The Netherlands.
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Kraaijevanger, J.F.B.M. Two counterexamples related to the Kreiss matrix theorem. BIT 34, 113–119 (1994). https://doi.org/10.1007/BF01935020
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DOI: https://doi.org/10.1007/BF01935020