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Visualizing Complex Eigenvalues of Real Matrices with 3D Plots

2025-07-21

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.

Geometric Intuition Behind the Inverse Function Theorem and Legendre Transformation

2025-05-05

This article provides a geometrically intuitive explanation of the inverse function theorem and Legendre transformation. Avoiding dry formula derivations, the author uses visual methods like graph transformations and reflections to illuminate the relationship between the derivative of an inverse function and its original function, and how the Legendre transformation solves integrals of inverse functions. Using arctan x as an example, the article clearly explains the application of these important mathematical tools, highlighting their broad use in fields like physics.