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Volume 2, Issue 4
Bounds on Condition Number of a Matrix

Hong-Ci Huang

J. Comp. Math., 2 (1984), pp. 356-360.

Published online: 1984-02

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  • 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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@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 = {

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 $A$. 

}, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/9671.html} }
TY - JOUR T1 - Bounds on Condition Number of a Matrix AU - Hong-Ci Huang JO - Journal of Computational Mathematics VL - 4 SP - 356 EP - 360 PY - 1984 DA - 1984/02 SN - 2 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/9671.html KW - AB -

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 $A$. 

Hong-Ci Huang. (1984). Bounds on Condition Number of a Matrix. Journal of Computational Mathematics. 2 (4). 356-360. doi:
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