Given some matrix $A$ of the form: $$ A \equiv \left(H^\mathrm{H}\Sigma H + \Lambda\right) $$ with $\Sigma$ and $\Lambda$ full-rank diagonal real matrices, and $H$ a large rank deficient not-necessarily square complex matrix.
I wish to estimate $A^{-1}$.
Now, I have good reason for thinking that $$ A^{-1} \approx \left(H^\mathrm{H}H + k I\right)^{-1} $$ which I can pre-compute and store, such that I have a potential mechanism for refining the estimate of $A^{-1}$ via a Neumann series as: $$ A^{-1} \approx_{\mathrm{better}} \sum_{n=0}^{N}\left[I - \left(H^\mathrm{H}H + k I\right)^{-1}\left(H^\mathrm{H}\Sigma H + \Lambda\right)\right]^{n} \left(H^\mathrm{H}H + k I\right)^{-1} $$ which could be great.
The problem I have is the above series estimation of $A^{-1}$ depends on the magnitude of all the eigenvalues of the bit in the square brackets being strictly less than $1$.
What I would ideally like is a mechanism for proving whether the eigenvalue requirement holds. I think I can relatively easily assert bounds on the eigenvalues, but I suspect they will be overly pessimistic in practice, and may overly limit the scope of the approach.
What I'm trying to get to is some method estimating whether the eigenvalues are in range. I'm thinking a strategy based on multiplying a white noise vector might give me the information I need (subject to uncertainty on expectations), but I don't have the theoretical justification for that.
Are there approaches I can use to estimate when the above series will converge, and better, whether it will converge quickly?
Of course, in the matrix-vector multiplication case, an approach is to use the initial estimate of $A^{-1}$ as a preconditioner for a CG algorithm, but that doesn't give me an estimate of the actual matrix (which is helpful), and also doesn't take account of structure I know a priori.
I'm also keen to consider other strategies if I'm missing something (decomposition approaches etc)!