A circle and a hyperbola living in one plot
We will see that the 3D plot of \(x^2 + (y + zi)^2 = 1\), where \(x\), \(y\), \(z\) are real and \(i\) is the imaginary unit, contains both a circle and a hyperbola. This visualization sheds light on the complex eigenvalues of real matrices. Let’s start by expanding the equation \(x^2+(y+zi)^2 = 1\) and separating it into real and imaginary parts. We get: \[\begin{align*} &\text{Real Part:} &x^2 + y^2 - z^2 &= 1, \\ &\text{Imaginary Part:} &yz &= 0. \end{align*}\] The condition \(yz=0\) split