If $\|\ f \|\ = \max_{|x|=1} |f(x)|$ then is $\|\ f \|\ \|\ f^{-1}\|\ = 1$ for all $f\in \mathcal{L}(\mathbb{R}^m,\mathbb{R}^n)$?

Question

Since $f: \mathbb{R}^m \to \mathbb{R}^n$ we can represent $A$ as the matrix of the linear transformation $f$, and may express the norm as $\|\ A \|\ $. Then, of course for the identity matrix, $I$, we have $\|\ I \|\ = 1$. And $AA^{-1} = I$ when $A$ has an inverse, so is it safe to say that $\|\ f \|\ \|\ f^{-1} \|\ = 1$ for every linear transformation $f$ where $f$ has an inverse. If not, what would we be a counter-example?

Answer 1

Take $f\colon\mathbb{R}^2\longrightarrow\mathbb{R}^2$ defined by $f(x,y)=(x,2y)$. Then $f^{-1}(x,y)=\left(x,\frac y2\right)$, $\|f\|=2$, and $\|f^{-1}\|=1$.