Unstable matrix equations and their relationship to measuring the emittance of an electron beam using beam position monitors

In general, the spatial distributions of electron beams from photoinjectors are unknown and are not well approximated by a Gaussian. Therefore, when measuring the emittance, it is important to make no assumptions about the beam's spatial distribution. An emittance diagnostic that fulfills this requirement uses beam position monitors to measure the second moment of the electron beam's image charge. This information, coupled with the beam line's transfer matrix, forms a set of linear equations. The least squares solution to these equations estimates the necessary beam parameters. More often than not, however, this solution is unstable and amplifies the errors in the second moment measurement. For this diagnostic to be viable, a stable implementation has to be found. Described here is an approach that accomplishes this task.