(untitled) (Journal of Computational Mathematics, 2021)
For each vector norm ‖x‖, a matirx A has its operator norm ‖A‖=min_x≠0‖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 AC^n× n, there is no finite upper bound of P(A) whch ‖·‖ varies on U if A eq α I; on the other hand, it is shown that inf_‖·‖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.