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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.
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. Let’s start by expanding the equation \(x^2+(y+zi)^2 = 1\) and separating it into real and imaginary parts. You can create these sort of plots for any other 2 by 2 matrix that depends on one real parameter using a desmos calculator I made.
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