I saw this post on the eigenvalues of a matrix plus a constant times the identity matrix.
Say $A$ is an $n\times n$ matrix (real and non-singular) with eigenvalues $\lambda_1,\ldots,\lambda_n$, then the eigenvalues of $A+cI$ are $\lambda_1+c,\ldots,\lambda_n+c$.
My question is on the condition number of $A+cI$, can this also be expressed in terms of the condition number of $A$? We have $$\kappa(A)=\sqrt{\frac{\lambda_{\mathrm{max}}\left(A^TA\right)}{\lambda_{\mathrm{min}}\left(A^TA\right)}},$$ so $$\kappa(A+cI)=\sqrt{\frac{\lambda_{\mathrm{max}}\Big((A+cI)^T(A+cI)\Big)}{\lambda_{\mathrm{min}}\Big((A+cI)^T(A+cI)\Big)}}.$$ Now of course $\lambda(A)=\lambda(A^T)$, but is there a relation between $\lambda(A)$ and $\lambda(A^TA)$? That is really all that I'm missing to connect $\kappa(A+cI)$ with $\kappa(A)$.
Edit I'll restate my question, as the current answer is not what I was looking for. I am not necessarily looking for a relation of the form $\kappa(A+cI)=f(\kappa(A))$ for some function $f$, since that apparently does not exist.
What I hope does exist, is a relation between $\lambda(A)$ and $\lambda(A^TA)$, such that I can rewrite the above and arrive at some relation between $\kappa(A)$ and $\kappa(A+cI)$. So, does that exist?
For example when $A$ is SPD, $$\kappa(A)=\frac{\lambda_{\mathrm{max}}(A)}{\lambda_{\mathrm{min}}(A)}\quad \mbox{ and }\quad \kappa(A+cI)=\frac{\lambda_{\mathrm{max}}(A)+c}{\lambda_{\mathrm{min}}(A)+c}$$ and $$\kappa(A)<\kappa(A+cI)\quad \mathrm{ or } \quad \kappa(A)>\kappa(A+cI) $$ depending on $c$. Now I was wondering if similar results could be derived for general $A$.