Let ∥A∥p,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 ∥A∥r,s/∥A∥p,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.
Bibliographical noteFunding Information:
by NSF Grant DMS-9424346.
- Marrix norm bounds
- Matrix inequalities