On matrices for which norm bounds are attained Original Research Article

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.