If I have a bound such that $ C \leq ||A^{-1}-B|| \leq D$, does it imply anything about on the $||B^{-1}-A||$, where A, B are square invertible, positive definite matrices with elements in R.
2026-04-02 20:08:04.1775160484
Relationship between $||A^{-1}-B||$ and $||B^{-1}-A||$
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$$B^{-1} - A = A(A^{-1} B^{-1} - I) = A(A^{-1} - B)B^{-1}$$
So if $\| \cdot \|$ is a submultiplicative norm, you get the following estimate: $$\|B^{-1} - A\| \le \|A\| \|A^{-1} - B\| \|B^{-1}\| \le \|A\| \|B^{-1}\| D.$$
Similarly, the inequality $\|A^{-1} - B\| \le \|B\|\|B^{-1} - A\| \|A^{-1}\|$ holds, which implies
$$\|B^{-1} - A\| \ge C \|B\|^{-1} \|A^{-1}\|^{-1}.$$
These estimates should be tight, just look at the case of $1 \times 1$-matrices.