Introduction In numerical linear algebra one meets condition numbers [[AII []A-11[ and similar quantities such as (max [a,i[)[[A-1H and [[Ai[[ [[A-l[], where A = (a,i) and A i is the /'-th column of A. The norms are very diverse. The problem then is to determine a rowand/or column-scaling of A which minimizes the quanti ty under consideration. I t is the purpose of this paper to specify a class of such quantities for which those scalings can be given explicitly. The results will be extensions of some results in [2]. They will also hold for non-square matrices. All proofs will be completely elementary. Also, in some cases where the minimizing scaling cannot be given explicitly, it can be said how far at most for a certain scaling the quanti ty under consideration may be away from its minimum.