@Article{JCM-2-356,
author = {Hong-Ci Huang},
title = {Bounds on Condition Number of a Matrix},
journal = {Journal of Computational Mathematics},
year = {1984},
volume = {2},
number = {4},
pages = {356--360},
abstract = {<p style="text-align: justify;">For each vector norm&nbsp;‖x‖, a matirx $A$ has its operator norm $‖A‖=\mathop{\rm min}\limits_{x≠0}\frac{‖Ax‖}{‖x‖}$&nbsp;and a condition number $P(A)=‖A‖&nbsp;‖A^{-1}‖$. 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&nbsp;$\mathop{\rm inf}\limits_{‖·‖\in U}&nbsp;‖A‖&nbsp;‖A^{-1}‖ =ρ(A)ρ(A^{-1})$ and in which case this infimum can or cannot be attained, where $ρ(A)$ denotes the spectral radius of $A$.&nbsp;</p>},
issn = {1991-7139},
doi = {https://doi.org/},
url = {http://global-sci.org/intro/article_detail/jcm/9671.html}
}