In the case when eigenvalues are complex we have to find real and imaginary parts of the complex-valued solutions which we obtain in the same way as in the real case.

Example 3. Solve .

Solution:

> A:=matrix(2,2,[-1/2,1,-1,-1/2]);

> E:=array(1..2,1..2,identity);

> t:=matrix(2,1,[t1,t2]);

> eigenvalues(A);

> multiply(A-(-1/2+I)*E,t);

We set t1=1, and solve -It1+t2=0 in t2.

> solve(-I+t2=0,t2);

Hence one eigenvector is and the corresponding solution is

. This is a complex-valued solution so we have to find the real and imaginary part. Rewrite as . Therefore we can write the complex solution in the form

. Hence two real and independent solutions are

and .