Shape of a matrix

I am using an evolutionary fitting method which provides the inverse Hessian up to a constant, i.e.

$\mathbf{C} = const \quad \mathbf{H}^{-1}$

The 2-norm of $\mathbf{C}$ is very small, e.g. $10^{-20}$, but the eigenvalues of $\mathbf{C}$ are relative to the eigenvalues of $\mathbf{H}$. I was reading a proof which had to do with scaling a matrix via eigenvalues in the form

$factor=\frac{\sum_j^p f(x) \lambda_j}{\sum_j ^p f(x)}$

I am working with log-likelihood, so $f(x)$ would have to be $\log[L(\beta)]$ in the above equation. Firstly, is there a known way to obtain $\mathbf{H}^{-1}$ by rescaling with the eigenvalues of $\mathbf{C}$, or do I need to calculate log-likehood for each record, sum them, and then solve for $\mathbf{H}^{-1}$?