  Let $\|A\|_{p,q}$ be the norm induced on the matrix $A$ with $n$ rows and $m$
columns by the H\"older $\ell_p$ and $\ell_q$ 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
$\|A\|_{r,s}/\|A\|_{p,q}$. 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.

Rings and Algebras