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On the condition number of a Kreiss matrix
- Part of
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- Journal:
- Canadian Mathematical Bulletin / Volume 66 / Issue 4 / December 2023
- Published online by Cambridge University Press:
- 29 May 2023, pp. 1376-1390
- Print publication:
- December 2023
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- Article
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In 2005, N. Nikolski proved among other things that for any
$r\in (0,1)$ and any 
$K\geq 1$, the condition number 
$CN(T)=\Vert T\Vert \cdot \Vert T^{-1}\Vert $ of any invertible n-dimensional complex Banach space operators T satisfying the Kreiss condition, with spectrum contained in 
$\left \{ r\leq |z|<1\right \}$, satisfies the inequality 
$CN(T)\leq CK(T)\Vert T \Vert n/r^{n}$ where 
$K(T)$ denotes the Kreiss constant of T and 
$C>0$ is an absolute constant. He also proved that for 
$r\ll 1/n,$ the latter bound is asymptotically sharp as 
$n\rightarrow \infty $. In this note, we prove that this bound is actually achieved by a family of explicit 
$n\times n$ Toeplitz matrices with arbitrary singleton spectrum 
$\{\lambda \}\subset \mathbb {D}\setminus \{0\}$ and uniformly bounded Kreiss constant. Independently, we exhibit a sequence of Jordan blocks with Kreiss constants tending to 
$\infty $ showing that Nikolski’s inequality is still asymptotically sharp as K and n go to 
$\infty $.
