# Summary

Due to some limitations, the work I’ve done this some was still fairly basic. So far, the research involves some computer programing and mathematical proofs. The coding part includes two Mathematica programs to numerically solve stable regions and Python program to run Monte Carlo trials. With the help of these programs, I looked into the stable region of system A-Pt where A and P are 3 by 3 real matrices when t progresses from negative infinity to positive infinity.  I found the possible number of times that the stability of the system could change and found substantial examples for each situation. Regarding the situations when the system’s stability changes for an odd number of times, some observations are also made and proved.  I also digressed a little bit to look into the probability that a random 3 by 3 real matrix is stable. I got an approximation with numerical experiment, but I haven’t started any theoretical proofs.

The summer is close to the end but the work is far from being done.  Following the current works, future directions will include some efforts to find the all possible configurations of stable regions and the corresponding conditions for each possibility. It will also be interesting to study the eigenvectors of the system. The three entries of eigenvector will form sign patterns (plus or minus.) For vector in R3, there exist four sign patterns (++-),(+-+),(-++) and (+++). To find the specific conditions for the each of the sign patterns to exist or change will be a good direction as well. These are the works to be done after the semester starts.