Inverse Efficiency Matrix (error)

Gold Member
Hi, let's say in some experiment with ##Z^0## (eg LEP) you are able to determine the "misidentification" of your particles.
Then you can find the efficiency matrix ##M_{eff}## which is given (for ##Z^0## decays to leptons or hadrons):

$\begin{pmatrix} N_e \\ N_\mu \\ N_\tau \\ N_{had} \end{pmatrix}_{detect} = M_{eff} \begin{pmatrix} N_e \\ N_\mu \\ N_\tau \\ N_{had} \end{pmatrix}_{real} \Rightarrow \vec{N}_{det} = M_{eff} \vec{N}_{real}$

With some errors in its elements, so in general you measure ##M_{eff} \pm \delta M_{eff}##.

My question is then, since I want to find what is actually my real measured events for the decay products:

$\vec{N}_{real} = M_{eff}^{-1} \vec{N}_{det}$

How can you find the error in the elements of $M_{eff}^{-1}$?

One example I thought of is quite primitive, and in general very untrustworthy, for example get $M_{eff}+\delta M_{eff}$ find its inverse, and do the same for $M_{eff} - \delta M_{eff}$. Then I know how much (in the worst case scenario the elements will diverge from the mean valued $M_{eff}^{-1}$.

I find it more plausible to create different matrices $A_i$ which have as elements different values in the range allowed from $M_{eff} \pm \delta M_{eff}$, take their inverse $A_i^{-1}$ and then mean value it and find its statistical error.
However with a ##4 \times 4## matrix (or 16 elements) I don't know how many $A_i$'s can give a trustworthy result...any help?

One example I thought of is quite primitive, and in general very untrustworthy, for example get $M_{eff}+\delta M_{eff}$ find its inverse, and do the same for $M_{eff} - \delta M_{eff}$. Then I know how much (in the worst case scenario the elements will diverge from the mean valued $M_{eff}^{-1}$.