Assuming that $$\|A\|=\max_{x\neq 0} \frac{\|Ax\|}{\|x\|}$$ how to prove the following? $$\|A^{-1}\|=\min_{x\neq 0} \left(\frac{\|Ax\|}{\|x\|}\right)^{-1}$$
This expression arises in page 55 of the book Scientific Computing by Michael Heath. I know that $$\|I\| \le \|A\| \cdot \|A^{-1}\|$$
In the supremum $$\|A\|:=\sup_{x\in\mathcal V-\{\mathbf 0\}}{\|Ax\|\over\|x\|}$$ $x$ is a variable over the non-zero members of some implicit vector space $\mathcal V$ on which $A$ acts as a linear automorphism. Therefore, we can make the substitution $y=Ax$ in the formula for $\|A^{-1}\|$ ; $$\|A^{-1}\|=\sup_{y\in\mathcal V-\{\mathbf 0\}}{\|A^{-1}y\|\over\|y\|}=\sup_{x\in\mathcal V-\{\mathbf 0\}}{\|x\|\over\|Ax\|}=[\inf_{x\in\mathcal V-\{\mathbf 0\}}{\|Ax\|\over\|x\|}]^{-1}$$