I'm studying the least squares problem. My professor told in class that small perturbation in $A$ does not mean small perturbation in $A^TA$, and so the relative perturbation of $x$ will depend on $\kappa^2(A),$ where $\kappa(A)$ is the condition number of $A$. I want to get an estimation of $\frac{\|\Delta x\|}{\|x\|},$ when we perturb $A$, but got stuck. Here's my attempt so far: $$A^TAx = A^Tb.$$ Perturbing $A$ gives us $(A-\Delta A)^T(A-\Delta A)(x-\Delta x) = (A-\Delta A)^Tb.$ This implies $$ (A^T-(\Delta A)^T)(A-\Delta A)(x - \Delta x) = A^Tb-(\Delta A)^Tb \iff \\ (A^TA - A^T\Delta A - (\Delta A)^TA - (\Delta A)^T(\Delta A))x - (A^TA - A^T\Delta A - (\Delta A)^TA - (\Delta A)^T(\Delta A))(\Delta x) = A^Tb-(\Delta A)^Tb \iff \\ (- A^T\Delta A - (\Delta A)^TA - (\Delta A)^T(\Delta A))x - (A^TA - A^T\Delta A - (\Delta A)^TA - (\Delta A)^T(\Delta A))(\Delta x) = -(\Delta A)^Tb\\ $$
How should I proceed?
