For an $n\times n$ diagonizable matrix $A$, is there a relation between the eigenvalues of $A$ and the eigenvalues of $A^TA$?

I ask this because I am looking into the relation between $A$ and $A+cI$, specifically their condition numbers:

$$\kappa(A)=\sqrt{\frac{\lambda_{\mathrm{max}}\left(A^TA\right)}{\lambda_{\mathrm{min}}\left(A^TA\right)}}\quad \mbox{ and }\quad \kappa(A+cI)=\sqrt{\frac{\lambda_{\mathrm{max}}\Big((A+cI)^T(A+cI)\Big)}{\lambda_{\mathrm{min}}\Big((A+cI)^T(A+cI)\Big)}}$$

I know for SPD matrices this reduces to

$$\kappa(A)=\frac{\lambda_{\mathrm{max}}(A)}{\lambda_{\mathrm{min}}(A)}\quad \mbox{ and }\quad \kappa(A+cI)=\frac{\lambda_{\mathrm{max}}(A)+c}{\lambda_{\mathrm{min}}(A)+c},$$ such that I can derive stuff like $$\kappa(A)<\kappa(A+cI)\quad \mathrm{ or } \quad \kappa(A)>\kappa(A+cI) $$ depending on $c$.

However, for general $A$ when I try rewriting $\kappa(A)$ I don't know what to do with the $\lambda(A^TA)$ part.

Can anyone help me? Or provide another trick to relate $\kappa(A)$ and $\kappa(A+cI)$ for general $A$, like I did for SPD matrices.