# On the sharpness of an inequality involving operator norm of invertible matrices

Let $$A\in GL_n(\mathbb R)$$.

How to show that for every $$\delta >0$$, $$\exists 0\ne r,b \in \mathbb R^n$$ such that $$\delta=||r||_2/||b||_2$$ and

$$\dfrac {||r||_2}{||A^{-1}||_2||A||_2||b||_2}=\dfrac{||A^{-1}r||_2}{||A^{-1}b||_2}$$ ?

NOTE: In general, I can prove that $$\dfrac {||r||_2}{||A^{-1}||_2||A||_2||b||_2}\le \dfrac{||A^{-1}r||_2}{||A^{-1}b||_2}, \forall r,b\in \mathbb R^n \setminus \{0\}$$ .

Just take $$r$$ as $$\delta$$ times the unit left singular vector for the largest singular value of $$A$$ and $$b$$ as the unit left singular vector for the smallest singular value of $$A$$.