1 Triple Eigenvalue with 3 Linearly Independent Eigenvectors
In this case we will have the eigenvalue l with the three linearly independent eigenvectors ?h1, ?h2 and ?h3 thus we find the following three linearly independent solutions,
elt ?h1 elt ?h2 elt ?h3
4 x 4 Systems
We will close this section out along with a couple of comments about 4x4 systems. In these cases we'll contain 4 eigenvalues and will require 4 linearly independent solutions in order to find a general solution. The vast majorities of the cases this time are natural extensions of what 3x3 systems cases and actually will use a vast majority of that work.
Now there are a couple of new cases which we must comment briefly on though. With 4x4 systems this will now be possible to contain two different sets of double eigen-values and two dissimilar sets of complex conjugate eigen-values. In either of these cases we can use each one as a separate case and utilize our previous knowledge regarding to double eigenvalues and complex eigenvalues to find the solutions we require.
This is also now possible to contain a "double" complex eigenvalue. Conversely we can have l = (a + bi) each occurs double in the list of eigenvalues. The solutions for that case aren't too bad. We find two solutions in the normal method of dealing along with complex eigenvalues. The rest two solutions will approach from the work we did for a double eigenvalue. The work we did there which case did not need such the eigenvalue/vector pair to be real. Thus if the eigenvector related withl = (a + bi) is then the second solutiuon will be as,
t e(a + bi) t ?h + e(a + bi )t ?r here ?r satisfies (A -lI) ?r = ?h
And once we've determined ?r we can again split it up in its real and imaginary parts by using Euler's formula to find two new real valued solutions.
At last with 4 x 4 systems we can now contain eigenvalues with multiplicity of 4. In these cases we can contain 1, 2, 3, or 4 linearly independent eigenvectors and we can utilize our work along with 3x3 systems to notice how to generate solutions for such cases. The one matter which you'll require to pay attention to is the conditions for the 2 and 3 eigenvector cases will have identical complications as the 2 eigenvector cases has in the 3 x 3 systems.
Therefore, we've discussed several of the matters involved in systems larger than 2 x 2 and this is hopefully clear that while we move into larger systems the work can be turn into vastly more complex.