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 \(yz=0\) splits into two cases:
\[\begin{align*} \text{Case 1: }y&=0 \text{ and so } x^2-z^2 = 1\\ \text{Case 2: }z&=0 \text{ and so } x^2+y^2 = 1 \end{align*}\]
Case 1 nets us a hyperbola in the \(xz\)-plane. Case 2 nets us a unit circle in the \(xy\)-plane.
You can check out the plot at desmos.
Eigenvalues and dynamical systems
Why does this matter beyond the visuals? These kinds of plots often appears when studying the (complex) eigenvalues of a real matrix that depends on a real parameter. For example, consider the matrix
\[M(\mu) = \begin{bmatrix} 0 & 1+\mu \\ 1-\mu & 0 \end{bmatrix}\]
... continue reading