On matrices for which norm bounds are attained

Let $\mathrm{\parallel }A{\mathrm{\parallel }}_{p,q}$ be the norm induced on the matrix $A$ with $n$ rows and $m$ columns by the H"older ${\mathrm{\ell }}_p$ and ${\mathrm{\ell }}_q$ norms on $R^n$ and $R^m$ (or $C^n$ and $C^m$), respectively. It is easy to find an upper bound for the ratio $\mathrm{\parallel }A{\mathrm{\parallel }}_{r,s}\mathrm{/}\mathrm{\parallel }A{\mathrm{\parallel }}_{p,q}$. 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.