Equal roots

If the characteristic polynomial of does not have distinct roots, then may not have linearly independent eigenvectors. For example, the matrix



has only two distinct eigenvalues and and two linearly independent eigenvectors, which we take to be

   

Consequently, the differential equation has only two linearly independent solutions



of the form . Our problem, in this case, is to find a third linearly independent solution. More generally, suppose that the matrix has only linearly independent eigenvectors. Then, the differential equation has only linearly independent solutions of the form . Our problem is to find an additional linearly independent solutions.

We approach this problem in the following ingenious manner. Recall that is a solution of the scalar differential equation , for every constant . Analogously, we would like to say that



is a solution of the vector differential equation

                                                                            (1)

for every constant vector . However, is not defined if is an matrix. This is not a serious difficulty, though. There is a very natural way of defining so that it resembles the scalar exponential .



We now show how to find linearly independent vectors for which the infinite series can be summed exactly.

(1) Find all the eigenvalues and eigenvectors of . If has linearly independent eigenvectors, then the differential equation has linearly independent solutions of the form . (Observe that the infinite series terminates after one term if is an eigenvector of with eigenvalue .)

(2) Suppose that has only linearly independent eigenvectors. Then, we have only linearly independent solutions of the form . To find additional solutions we pick an eigenvalue of and find all vectors for which

,       but      
For each such vector ,



is an additional solution of . We do this for all the eigenvalues of .

(3) If we still do not have enough solutions, then we find all vectors for which

,       but      

For each such vector ,



is an additional solution of .

(4) We keep proceeding in this manner until, hopefully, we obtain linearly independent solutions.



EXAMPLE: Solve the system

                                                                 (3)



EXAMPLE: Find three linearly independent solutions of the differential equation




EXAMPLE: Solve the initial-value problem