The definition of a Lyapunov exponent can be extended to include an imaginary part. This extension requires the definition of a coordinate frame on the tangent space of the differential equation and an extension of the concept of a limit. The definition of extended Lyapunov exponents is based on the eigenvalues of the fundamental matrix. It is shown that the extended exponent agrees completely with the constant-coefficient case. It is shown that the eigenvectors and eigenvalues obey differential equations and can be propagated numerically without constructing the fundamental matrix itself. Bifurcation of eigenvalues and eigenvectors can also be followed numerically without recourse to the fundamental matrix. Two example applications of the method to the calculation of extended Lyapunov exponents are given. In the Lorenz problem, the real parts of the extended Lyapunov exponents agree quite well with previous results. Fourier-transform methods are used to show that the power spectrum of relative motion is discrete, with fundamental frequency quite close to the calculated imaginary part of the extended Lyapunov exponent. In the simple pendulum, the extended Lyapunov exponents are usually purely imaginary and are the relative oscillation frequencies of adjacent trajectories.