Complex Eigenvalues 1. Complex Eigenvalues

In the previous note, we obtained the solutions to a homogeneous linear system with constant coefficients. x = A x under the assumption that the roots of its characteristic equation |A − λI| = 0, — i.e., the eigenvalues of A — were real and distinct. In this section we consider what to do if there are complex eigenval­ ues. Since the characteristic equation has real coefficients, its complex roots must occur in conjugate pairs: λ = a + bi, λ ¯ = a − bi. Let's start with the eigenvalue a + bi. According to the solution method described in the note Eigenvectors and Eigenvalues, (from earlier in this ses­ sion) the next step would be to find the corresponding eigenvector v, by solving the equations (a − λ)a 1 + ba 2 = 0 ca 1 + (d − λ)a 2 = 0 for its components a 1 and a 2. Since λ is complex, the a i will also be com­ plex, and therefore the eigenvector v corresponding to λ will have complex components. Putting together the eigenvalue and eigenvector gives us for­ mally the complex solution x = e (a+bi)t v. (1) Naturally, we want real solutions to the system, since it was real to start with. To get them, the following theorem tells us to just take the real and imaginary parts of (1). (This theorem is exactly analogous to what we did with ordinary differential equations.). Theorem. Given a system x = Ax, where A is a real matrix. If x = x 1 + i x 2 is a complex solution, then its real and imaginary parts x 1 , x 2 are also solutions to the system.