I'm look at this tiny book called "Deblurring Images: Matrices, Spectra, and Filtering" by Hansen, Nagy, O'Leary.
This is a self study, but I believe my question is broad enough so that it can be of benefit to the math community.
My question stems from page 7 of this book:
Given $B = A_cXA_r^T + E$, show that the relative error in $X' = A_c^{-1}BA_r^{-T}$ satisfies
$\frac{||X'-X||_F}{||X||_F} \leq K(A_c)K(A_r)\frac{||E||_F}{||B||_F}$
where $K(A_c) = ||A_c||||A_c^{-1}||$ is the condition number of $A_c$. I can't seem to get the exact bound but perhaps someone can correct what I am doing wrong, that would be great. From now on, the norm i'll be using is the Frobenius norm.
Here's what I did:
$||X'-X|| = ||A_c^{-1}EA_r^{-T}|| \leq ||A_c^{-1}||\cdot ||E|| \cdot ||A_r^{-T}|| = ||A_c^{-1}||\cdot ||E|| \cdot ||A_r^{-1}|| = \frac{K(A_c)K(A_r)||E||}{||A_c||\cdot ||A_r||}$
It follows that
$\frac{||X'-X||}{||X||} \leq \frac{K(A_c)K(A_r)||E||}{||X||\cdot ||A_c|| \cdot ||A_r||} \leq \frac{K(A_c)K(A_r)||E||}{||A_cXA_r^T||} = \frac{K(A_c)K(A_r)||E||}{||B-E||}$
Though the denominator isn't quite what I am hoping for. Is there something I am missing?