Matrix norm for which $\|A\| \|A^{-1}\| \leq c$?

Question

Is there any matrix norm on $\mathbb{R}^{n \times n}$ such that $\|A\| \|A^{-1}\| \leq c$ for all invertible matrices $A$?

The more general inequality $\|A\| \|B\| \leq c \| A B\|$ cannot hold because there are non-zero matrices whose product is zero.

Answer 1

If you want $\|A\|\|A^{-1}\|$ to be bounded on $GL_n(\mathbb R)$, this is impossible unless $n=1$. The reason is simple: we have $\rho(A)\rho(A^{-1})\le\|A\|\|A^{-1}\|$ but $\rho(A)\rho(A^{-1})$ is not bounded above. Consider $A=\operatorname{diag}(k,\frac1k,1,1,\ldots,1)$ for instance.

Alternatively, since all norms are equivalent on a finite-dimensional vector space, if $\|A\|\|A^{-1}\|$ is bounded for any particular norm, it must be bounded for all norms. However, $\|A\|\|A^{-1}\|$ is unbounded when $A=\operatorname{diag}(k,\frac1k,1,1,\ldots,1)$ and the Frobenius norm is used. Hence $\|A\|\|A^{-1}\|$ is also unbounded for all other norms.