Year: 1984
Journal of Computational Mathematics, Vol. 2 (1984), Iss. 4 : pp. 356–360
Abstract
For each vector norm ‖x‖, a matirx $A$ has its operator norm $‖A‖=\mathop{\rm min}\limits_{x≠0}\frac{‖Ax‖}{‖x‖}$ and a condition number $P(A)=‖A‖ ‖A^{-1}‖$. Let $U$ be the set of the whole of norms defined on $C^n$. It is shown that for a nonsingular matrix $A\in C^{n\times n}$, there is no finite upper bound of $P(A)$ whch ‖·‖ varies on $U$ if $A\neq \alpha I$; on the other hand, it is shown that $\mathop{\rm inf}\limits_{‖·‖\in U} ‖A‖ ‖A^{-1}‖ =ρ(A)ρ(A^{-1})$ and in which case this infimum can or cannot be attained, where $ρ(A)$ denotes the spectral radius of $A$.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/1984-JCM-9671
Journal of Computational Mathematics, Vol. 2 (1984), Iss. 4 : pp. 356–360
Published online: 1984-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 5