On matrices for which norm bounds are attained

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Let Aₚ,q be the norm induced on the matrix A with n rows and m columns by the Hölder ℓₚ and ℓq norms on Rⁿ and Rᵐ (or Cⁿ and Cᵐ), respectively. It is easy to find an upper bound for the ratio Aᵣ,ₛ/Aₚ,q. In this paper we study the classes of matrices for which the upper bound is attained. We shall show that for fixed A, attainment of the bound depends only on the signs of r-p and s-q. Various criteria depending on these signs are obtained. For the special case p=q=2, the set of all matrices for which the bound is attained is generated by means of singular value decompositions.