MATH 374 Lecture 23 Complex Eigenvalues

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].

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)

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

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

Complex Eigenvalue Case

Complex Eigenvalue Case

Complex Eigenvalue Case

Complex Eigenvalue Case

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).

Example 1

Example 1

Example 1

Example 1