I have a matrix of the form $\tilde{H} = H + i A A^\dagger$. It is known that $H$ is hermitian and that $\tilde{H}$ is invertible and $A A^\dagger$ has a kernel of dimension $\geq 1$. I want to study the effect that adding $i A A^\dagger$ to $H$ has on the condition number of the resulting matrix as a function of the ratio $\left \| A A^\dagger \right \| / \left \| H \right \|$ and in terms of the condition number $K(H)$ of $H$. Some kind of an upper bound on the ratio $K(\tilde{H}) / K(H)$, at least for the cases $\left \| A A^\dagger \right \| \ll \left \| H \right \|$ and $\left \| A A^\dagger \right \| \gg \left \| H \right \|$ would be great.
What I have done: so far, I have been working in the $2$-norm. As $H$ is hermitian, its condition number is the largest absolute eigenvalue divided by the smallest. My idea was for the case $\left \| A A^\dagger \right \| \ll \left \| H \right \|$ to regard $i A A^\dagger$ as a perturbation to estimate the maximum change in the smallest eigenvalue of $\tilde{H}$ w.r.t. the smallest of $H$. Applying perturbation theory yields that to first order, $\tilde{E}_{\text{min}} \leq \left | E_{\text{min}} + i \left \| A A^\dagger \right \| \right |$. Unfortunately, as $\tilde{H}$ is not hermitian anymore, I am not sure if the smallest eigenvalue remains a significant quantity (as for non-hermitian matrices, the singular values are the ones of importance).
For the case $\left \| A A^\dagger \right \| \gg \left \| H \right \|$ the only thing I was able to predict and verify experimentally is that the condition number grows almost linearly with $\left \| A A^\dagger \right \|$ as the largest eigenvalue grows, too.
I am really stuck and have run out of ideas. Any help is greatly appreciated.