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

The 2-norm of is very small, e.g. , but the eigenvalues of are relative to the eigenvalues of . I was reading a proof which had to do with scaling a matrix via eigenvalues in the form

I am working with log-likelihood, so would have to be in the above equation. Firstly, is there a known way to obtain by rescaling with the eigenvalues of , or do I need to calculate log-likehood for each record, sum them, and then solve for ?