# How to Invert a Poorly Conditioned Matrix

In my research, I need to invert a Fisher matrix in order to get a covariance matrix for me to do parameter estimation. Unfortunately, the values of Fisher matrix vary by many orders of magnitude, and the numerical errors are significant (see image at end).

I first attempted this in python using scipy.linalg.inv, and it is not giving stable enough results. Do you guys have any recommendations to proceed?

Edit: Some context for the comment. I am trying to find parameter uncertainties $$\theta^i$$ given measurement $$h(f,\theta^i)$$. The fisher matrix is defined as $$\Gamma_{ij} = \left( \frac{\partial h}{\partial \theta^i} | \frac{\partial h}{\partial \theta^j}\right) = 2 \int_0^\infty S_n(f)^{-1} \left(\frac{\partial h^\star}{\partial \theta^i} \frac{\partial h}{\partial \theta^j} + \frac{\partial h}{\partial \theta^i} \frac{\partial h^\star}{\partial \theta^i} \right) df \, .$$ Then the parameter uncertainties are $$\langle \Delta \theta^i \Delta \theta^j\rangle = \left( \Gamma^{-1} \right)^{ij} \, .$$ So my numerical issues are occuring when I try and find $$\left( \Gamma^{-1} \right)^{ij}$$.

• Inverting a matrix (espcially if large and badly scaled) is never a good idea. What do you need this matrix for? Please provide further context so that others can contribute – VoB Jan 22 at 10:07
• @VoB I added more context for the matrix I need to invert. – Brian.C.Seymour Jan 22 at 18:00

There is no simple fix. For an ill-conditioned matrix $$A$$, the harm (loss of precision) is already done the moment you wrote those numbers in a numpy array, because that tiny $$10^{-16}$$ perturbation from the exact non-representable values is already harmful.
You could increase your working precision; but at that point the question is if your matrix entries $$A_{ij}$$ can really be computed with more than 16 correct digits; the answer is almost surely no, for data that depend on real-world measurements.
Another hope is that a diagonal rescaling can improve the condition number; it looks like row and column 2 have the largest values in your data, so you could scale those down. This may give you better accuracy in single entries (but not necessarily if you are measuring the accuracy of the computed $$B \approx A^{-1}$$ with $$\|B-A^{-1}\|$$).