How can we show that the matrix differential equation dX/dt=AX+XB has the solution and prove that the solutions of dX/dt=AX-XA keep the same eigenvalues for all time.
Thanks in advance
You can check that that is indeed a solution by differentiating, now suppose we have a solution we will prove it has to be of that form (uniqueness)
Remember that the product rule also holds for Matrixes, we have:
and commute, for every matrix .
So it is a constant matrix ( because of the 0 derivative) :
Let we get:
And the uniqueness is proven.
For the second part of the question, let
The solutions is of the form:
Let's see the characteristic polynomial: