Let and be matrices as defined in this previous exercise, with . Compute the fundamental matrix of this differential equation where is one of the following matrices:
Let be a matrix, and let be in Jordan canonical form. Using this previous exercise, we have that is a fundamental matrix, and by this previous exercise, is a fundamental matrix of our differential equation. We can use this previous result (and part 2 of this previous exercise) to compute .
We computed the Jordan canonical forms (and conjugators) for these three matrices in this previous exercise.
For the matrix , we have and . Now , and . We can easily verify that this is indeed a fundamental matrix.
For the matrix , as computed in this previous exercise, if we let then is in Jordan canonical form. Now . So is a fundamental matrix.
Finally, consider . Letting (Seethis previous exercise), we have in Jordan canonical form. Then , and hence is a fundamental matrix.