Abstract
Let F be the field of real or complex numbers, and let G be a subgroup of the general linear group GL n( F ) . If A is an m× n matrix over F , let ∥·∥ A,∞ be the seminorm on F n , defined by ∥x∥ A,∞=∥Ax∥ ∞ for all x∈ F n. 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∈ F n and all S∈ G. As a special case we describe all matrices A for which ∥·∥ A,∞ is absolute or a symmetric gauge function.
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