Matrix Perturbation Theory

I plan on studying how small perturbations in a matrix can affect properties of the matrix. This has many applications, like in physics. Even modern physics such as M-theory uses matrices and perturbation techniques [3]. Many other physical systems many physical problems can be modeled with systems of differential equations. Many of these systems are linear systems, and if not, can be approximated by linear systems. At this point, the model is reduced down to something like