Length of inverse-matrix vector product.

Question

Assume that I have an invertible matrix $A$ and a vector $v$. Let $\|Av\| = k$. Can we deduce the value of $\|A^{-1}v\|$? My guts tell me that $\|A^{-1}v\|$ = $\frac{1}{k}$, but I haven't been able to prove it.

Answer 1

Let $A$ be the identity matrix for example. You would have $\|v\|=k$ and your gut feeling would tell you $\|v\|=1/k$.

$k$ and $1/k$ are not equal (in general).

Another example would be a rotation matrix, this won't change the length of $v$, the inverse matrix won't change the length either.

Answer 2

Take $A=\operatorname{diag} (1,2)$, $v=(1,1)^T$, $w=(0,\sqrt{5})^T$, then $\|Av\| = \|Aw\| = \sqrt{5}$, but $\|A^{-1} v\| = \sqrt{5 \over 4}$, $\|A^{-1}w\| = \sqrt{5}$.