It is a commonly occurring problem to find “good” norms ‖·‖ or logarithmic norms μ(·) for a given matrix in the sense that they should be close to respectively the spectral radius ρ(A) and the spectral abscissa α(A). Examples may be the certification thatA is convergent, i.e. ρ(A)≤‖A‖<1 or stable, i.e. α(A)≤μ(A)<0. Often the ordinary norms do not suffice and one would like to try simple modifications of them such as using an ordinary norm for a diagonally transformed matrix. This paper treats this problem for some of the ordinary norms.