Prove that $\frac{1}{K(B)}\frac{\|A-B\|}{\|A\|} \le \frac{\|A^{-1}-B^{-1}\|}{\|B^{-1}\|} \le K(A)\frac{\|B-A\|}{\|A\|}.$

Question

Let $A$ and $B$ be invertible matrices and $K(C)$ the condition number of an invertible matrix $C$. Prove that $$\frac{1}{K(B)}\frac{\|A-B\|}{\|A\|} \le \frac{\|A^{-1}-B^{-1}\|}{\|B^{-1}\|} \le K(A)\frac{\|B-A\|}{\|A\|}.$$

I tried \begin{align} \frac{1}{K(B)}\frac{\|A-B\|}{\|A\|} &\le \frac{\|A^{-1}-B^{-1}\|}{\|B^{-1}\|} \le K(A)\frac{\|B-A\|}{\|A\|} &&\Leftrightarrow \\[5pt] \frac{1}{K(B)}\frac{\|A\|-\|B\|}{\|A\|} &\le \frac{\|A^{-1}\|-\|B^{-1}\|}{\|B^{-1}\|} \le K(A)\frac{\|B\|-\|A\|}{\|A\|} &&\Leftrightarrow \\[5pt] \frac{\|A-B\|}{\|A\|\|B\|\|B^{-1}\|} &\le \frac{\|A^{-1}\|}{\|B^{-1}\|}\le \frac{\|A\|\|A^{-1}\|(\|B\|-\|A\|)}{\|A\|} &&\Leftrightarrow \\[5pt] \frac{\|A-B\|}{\|A\|\|B\|} &\le \|A^{-1}\|\le \frac{\|A^{-1}\|(\|B\|-\|A\|)}{\|A\|} &&\Leftrightarrow \\[5pt] \frac{\|A-B\|}{\|B\|} &\le K(A)\le \|A^{-1}\|(\|B\|-\|A\|) &&\Leftrightarrow \\[5pt] \frac{\|A-B\|}{\|B\|} &\le K(A)\le \|A^{-1}\|\|B\|-K(A) &&\Leftrightarrow \\[5pt] \frac{\|A-B\|}{\|B\|} &\le 0 \le \|A^{-1}\|\|B\|-2K(A) &&\Leftrightarrow \\[5pt] \|A-B\| &\le 0 \le \|A^{-1}\|\|B\|-2K(A) &&\Leftrightarrow \\[5pt] \|A-B\| &\le 0 \le \|B- 2A\| \end{align}

I don't know how to continue. It doesn't look correct.

Answer 1

I don't know why are you doing $$||A-B|| = ||A|| - ||B||$$ at certain places because it's certainly not true.

Start with the condition number of $A$ and write the RHS of your inequality as: $$k(A) = \|A\|\|A^{-1}\|\geq\dfrac{\|A\|\|B^{-1} - A^{-1}\|}{\|B^{-1}\|\|B-A\|}$$ can you take it from here?

You would need to use that fact that operator norms are sub-multiplicative: $$\|XY\|\leq\|X\|\|Y\|$$

Answer 2

Can you prove that $\lVert A - B\rVert \leq \lVert A\rVert \lVert B\rVert\lVert A^{-1} - B^{-1}\rVert$? Then the first inequality is easy. Do the same with $\lVert A^{-1} - B^{-1}\rVert \leq \lVert A^{-1}\rVert\lVert B^{-1}\rVert\lVert A - B\rVert.$ for the other inequality.

Here we use the submultiplicativity property of matrix norms, i.e. $$ \lVert AB\rVert \leq \lVert A\rVert \lVert B\rVert, $$ which is not a property possessed by all norms (only induced and the Frobenius norm as far as I am aware).