Hollins Digital Commons

Real and Complex Eigenvalues

Glass Dining Room

An undergraduate course in Linear Algebra introduces eigenvalues and defines an eigenvalue as a number lamba so that Ax = (lamba)x, where A is an n×n matrix and x is a vector with n entries. The purpose of this project is to learn more about eigenvalues and discover the differences and similarities between the real and complex cases. I will investigate discrete dynamical systems and systems of first order linear differential equations. In these settings, real and complex eigenvalues produce predictable patterns. The predator prey problem is an application of eigenvalues and discrete dynamical systems to biology. In the discrete dynamical system setting, phase shift diagrams for a matrix with real eigenvalues create an attracting or saddle point and the phase shifts exhibit a spiraling pattern if the corresponding matrix has complex eigenvalues. Systems of first order linear differential equations are commonly used when studying physics. The graph of solution curves for a given system of first order linear differential equation shows an attracting, repelling, saddle, or spiraling behavior, based on whether the eigenvalues of the corresponding matrix are real or complex. It is interesting to see that the patterns for real vs. complex are similar in these two settings.