Suppose $C = B + \mathrm{i} c AA^\dagger$, where $^\dagger$ denotes the conjugate-transpose of a matrix. Here $c > 0$ is a free parameter, $B$ is hermitian and $AA^\dagger$ is obviously positive semi-definite.
I am interested in the first-order correction to the singular values of $C$, i.e. in the eigenvalues of $\sqrt{C^\dagger C}$ in the limit cases $c \rightarrow 0$ and $c \rightarrow \infty$ as a function of the properties of $B$ and $AA^\dagger$.
I have done most of the calculations using first order (non)-degenerate perturbation theory on the eigenvalues of $C$ and was able to give and estimate of the condition number of $C$ (in the spectral norm) in these limiting cases as a function of the eigenvalues and vectors of $B$ and $AA^\dagger$. So I simply pretended as if $B$ and $\mathrm{i} c AA^\dagger$ remain (skew)-hermitian under the effects of the perturbation in which case the absolute values of the eigenvalues are the significant quantities.
Interestingly, this seems to be in agreement with numerical experiments as I was able to confirm my calculations under the assumption that hermiticity is preserved.
Now I am wondering if it is possible to show that the first order correction on the singular values is (in norm) equal to that of the eigenvalues in this particular case and in general (and under which circumstances this is valid).
I'd be grateful for any help.