The isometries and the G-invariance of certain seminorms

AbstractLet 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∥∞forallx∈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