I am reading Kress's Numerical Analysis. There is a paragraph on page 84 which is discussing what will happen if we "perturb" an equation $Ax=y$. It says:
If for some $\delta\in \Bbb{C}$ we perturb the right-hand side by setting $y^{\delta}=y+\delta v_j$, we obtain a perturbed solution $x^{\delta}=x+\delta u_j/\mu_j$. Hence, the ratio $||x^{\delta}-x||_2/||y^{\delta}-y||_2=1/\mu_j$ becomes large if $A$ possesses small singular values.
where $\mu_1, ..., u_r$ are singular values of $A$, $A$ has a singular value decomposition $VDU^*$, $V=(v_1, ..., v_m)$ and $U=(u_1, ..., u_n)$.
My Quesiton: The definition $y^{\delta}=y+\delta v_j$ means I can choose any one of $v_1, ..., v_m$ to define $y^{\delta}$? Or I have to choose some $j$ which satisfies some condition? Like $||y^{\delta}-y||_2\leq \delta$?