Locally compatible generalized matrix norms

Summary

We show that the following three conditions are equivalent for a generalized matrix normN onM _k (C):

• (i) for eachx∈C ^k there is a vector norm · _x (depending onx) such that for allA∈M _k (C), Ax _x ≦N(A) x _x;

• (ii) for eachA∈M _k (C) there is a vector norm · _A (depending onA)_such that for allx∈C ^k,Ax _A ≦N(A) x _A; and

• (iii) the spectral radius ϱ(A)≦N(A) for allA∈M _k (C).

We call generalized matrix norms satisfying (i) (and, thus, (ii) and (iii)locally compatible and cite examples, for instance, to show that a locally compatible generalized matrix norm is not necessarily globally compatible with any single vector norm.