Monotonicity properties of certain classes of norms Original Research Article

Let p be a norm on Kn, where K = R or K = C. If S ε Kn,n is a nonsingular matrix, let ps be the norm on Kn, defined by pS(x) = p(Sx) for all x ε Kn. This note gives some conditions on S for which pS has a certain monotonicity property, and characterizes p by monotonicity properties of all norms pS with S in a given group of unitary (orthogonal) matrices. In particular, we obtain the following characterizations: (1) If p is an lq-norm, 1 < q < ∞, all matrices S are described for which pS is weakly monotonic. (2) If for each unitary (orthogonal) matrix S ε Kn,n the norm pS is quasimonotonic, then p is a positive multiple of the l2-norm.