The condition number
${c}_{\phi}$
of a nonsingular matrix
$A$
is defined by
${c}_{\phi}\left(A\right)=\phi \left(A\right)\phi \left({A}^{1}\right)$
where ordinarily
$\phi $
is a norm. It was proved by O. TausskyTodd that
$\left(c\right)$
${c}_{\phi}\left(A\right)\leqq {c}_{\phi}\left(A{A}^{\ast}\right)$ when
$\phi \left(A\right)={\left(trA{A}^{\ast}\right)}^{1\u22152}$ and
when
$\phi \left(A\right)$
is the maximum absolute characteristic root of
$A$. It is shown that
$\left(c\right)$ holds whenever
$\phi $ is a unitarily invariant
norm, i.e., whenever
$\phi $
satisfies
$\phi \left(A\right)>0$
for
$A\ne 0$;
$\phi \left(\alpha A\right)=\left\alpha \right\phi \left(A\right)$ for
complex
$\alpha $;
$\phi \left(A+B\right)\leqq \phi \left(A\right)+\phi \left(B\right)$;
$\phi \left(A\right)=\phi \left(AU\right)=\phi \left(AU\right)$ for all unitary
$U$. If in addition,
$\phi \left({E}_{ij}\right)=1$, where
${E}_{ij}$ is the matrix with one
in the
$\left(i,j\right)$th place and
zeros elsewhere, then
${c}_{\phi}\left(A\right)\geqq {\left[{c}_{\phi}\left(A{A}^{\ast}\right)\right]}^{1\u22152}$.
Generalizations are obtained by exploiting the relation between unitarily invariant
norms and symmetric gauge functions. However, it is shown that (c) is independent
of the usual norm axioms.
