1. MATH 374 Lecture 23
Complex Eigenvalues

2. 8.5: Complex Eigenvalues
Definition: For any complex number z = a+ib, with a and b real numbers, Re(z) = a, Im(z) = b, and the conjugate of z is the complex number
Similar definitions hold for complex matrices.
For any matrix A = [aij], Ā is the conjugate matrix [āij].

3. Homogeneous Systems Revisited
Consider again the system of n linear homogeneous equations in n unknowns:
X’ = AX. (1)
Looking for non-trivial solutions of (1) of the form X = Cemt, we find that m must be an eigenvalue of A with corresponding eigenvector C.
Recall that m is a root of the characteristic equation
|A – mI| = 0 (2)
and C is a non-zero solution of
(A – mI)C = 0. (3)

4. Complex Eigenvalue Case
Suppose for  and  real numbers, m =  + i is an eigenvalue of matrix A with corresponding eigenvector C.

5. Complex Eigenvalue Case
X’ = AX. (1)
(A – mI)C = 0. (3)
Complex Eigenvalue Case

6. Complex Eigenvalue Case

7. Complex Eigenvalue Case

8. Complex Eigenvalue Case

9. Complex Eigenvalue Case

10. Complex Eigenvalue Case
X’ = AX (1)
Complex Eigenvalue Case
Theorem 8.8: Let m =  + i  be an eigenvalue of the constant real-valued matrix A in (1) with corresponding eigenvector C.
Then
X1(t) = [Re(C) cost – Im(C) sint]et
and
X2(t) = [Im(C) cost + Re(C) sint]et
are linearly independent solutions of (1).

11. Example 1

12. Example 1

13. Example 1

14. Example 1