Why is $\kappa(A^{-1})= \kappa(A)$? In my syllabus we have the alternative definition of the condition of a matrix:
$$\kappa(A)= \frac{\text{max}_{\| \vec{y} \| =1}\| A \vec{y} \|}{\text{min}_{\| \vec{y} \| =1}\| A \vec{y} \|}$$
In it, it also says that by definition of the condition of a matrix it follows that $\kappa(A^{-1})= \kappa(A)$. So there is no explanation for this. Therefore, my question is: Why is $\kappa(A^{-1})= \kappa(A)$?

For an invertable matrix, your numerator and denominator are the absolute values of the largest and smallest eigenvalues. The eigenvalues of the inverse matrix are the reciprocals of its eigenvalues.
