Matrix Differential Equation - Solved Example of A Matrix ODE - Second Step

Second Step

As it was already mentioned above, in a simple description, this step involves finding the eigenvectors by means of using the information originally given to us.

For each of the eigenvalues calculated we are going to have an individual eigenvector. For our first eigenvalue, which is, we have the following:

Simplifying the above expression by applying basic matrix multiplication rules we have:

.

All of these calculations have been done only to obtain the last expression, which in our case is . Now taking some arbitrary value, presumably a small insignificant value, which is much easier to work with, for either or (in most cases it does not really matter), we substitute it into . Doing so produces a very simple vector, which is the required eigenvector for this particular eigenvalue. In our case, we pick, which, in turn determines that and, using the standard vector notation, our vector looks like this:

Performing the same operation using the second eigenvalue we calculated, which is, we obtain our second eigenvector. The process of working out this vector is not shown, but the final result is as follows:

Once we've found both needed vectors, we start the third and last step. Don't forget that we'll substitute the eigenvalues and eigenvectors determined above into a specialized equation (shown shortly).