Visualizing Complex Eigenvalues of Real Matrices with 3D Plots
This article explores the 3D plot of the equation x²+(y+zi)²=1 (where x, y, z are real numbers and i is the imaginary unit), revealing a circle and a hyperbola. Separating the equation into real and imaginary parts yields two cases: when y=0, x²-z²=1 (a hyperbola); when z=0, x²+y²=1 (a unit circle). This visualization offers insights into the behavior of complex eigenvalues of real matrices that depend on a real parameter. Two examples of 2x2 matrices are provided, demonstrating how this method analyzes eigenvalues. The article concludes by suggesting that this approach can be extended to other 2x2 matrices dependent on a single real parameter.