1. Solutions to Systems Of Linear Equations II
Multiple eigenvalues

2. What if l is an eigenvalue of multiplicity > 1?
So far we know how to treat distinct eigenvalues to get a general solution to y’=Ay
If there is a multiple eigenvalue (same term is squared in the factorization), a couple things can happen

3. Case 1 – 2 free variables Example
Let’s again get the eigenvalues for A, from the NSpire

4. Not a serious problem in this case
Thus x2 and x3 are both free, so we can make two choices for each, making sure they are linearly independent and that x1 conforms to the form of the base variable

5. We proceed as before
So l =6 (and thus e6t) gets two members of the fundamental set, one with v1 and one with v2
To get the third member of the fundamental set, get the eigenvector that corresponds to l = 0

6. Write the general solution as usual

7. One more case for double l
Sec 10.5

8. Here there isn’t any trickery to be done
There are two free variables to “switch”, so we need to do something else
Hint: It’s similar to our trick for the multiple roots for the characteristic polynomial for the first order linear w/ constant coefficients, but requires an extra term.