The link between the infinity norm of the difference between matrices and the infinity norm of the difference between these matrices inverted.

Question

I am trying to develop an inequality and I have a bit of difficulty... The problem is the following one,

$\underline{\text{notations}}$

• If $A$ is a $n \times n$ matrix with entries $a_{i,j}$, then $|A|_{\infty}$ = $\text{max}_{i,j} |a_{i,j}|$
• If $A$ is a $n \times n$ matrix, then $A^{-1}$ is the inverse of $A$.

$\underline{\text{problem}}$

We know that:

• $A$ and $B$ are two positive definite matrices
• $c$ is a strictly positive constant
• $|A-B|_{\infty}$ $\leq$ $c$

From this information, is it possible to say something about the upper bound of $|A^{-1}-B^{-1}|_{\infty}$?

Answer 1

No. Suppose that $A_n=\frac1n\operatorname{Id}$ and that $B=\operatorname{Id}$. Then $(\forall n\in\Bbb N):\|A_n-B\|_\infty\leqslant1$. However,$$\left\|A^{-1}-B^{-1}\right\|_\infty=\|n\operatorname{Id}-\operatorname{Id}\|_\infty,$$which can be arbitrarily large.