The isometries and the G-invariance of certain seminorms

Let F be the field of real or complex numbers, and let G be a subgroup of the general linear group GLn(F). If A is an m×n matrix over F, let • A,∞ be the seminorm on Fn, defined by x A,∞= Ax ∞ forall x Fn. In this paper we characterize the linear isometries for the seminorm • A,∞ and study the conditions on A for which • A,∞ is G-invariant; that is, Sx A,∞= x A,∞ for all x Fn and all S G. As a special case we describe all matrices A for which • A,∞ is absolute or a symmetric gauge function.