From Random Polygon to Ellipse: An Eigenanalysis

Abstract

Suppose x and y are unit 2-norm n-vectors whose components sum to zero. Let ${\cal P}(x,y)$ be the polygon obtained by connecting $(x_{1},y_{1}),\ldots,(x_{n},y_{n}),(x_{1},y_{1})$ in order. We say that $\widehat{{\cal P}}(\widehat{x},\widehat{y})$ is the normalized average of ${\cal P}(x,y)$ if it is obtained by connecting the midpoints of its edges and then normalizing the resulting vertex vectors $\widehat{x}$ and $\widehat{y}$ so that they have unit 2-norm. If this process is repeated starting with ${\cal P}_{0} = {\cal P}(x^{(0)},y^{(0)})$, then in the limit the vertices of the polygon iterates ${\cal P}(x^{(k)},y^{(k)})$ converge to an ellipse ${\cal E}$ that is centered at the origin and whose semiaxes are tilted forty-five degrees from the coordinate axes. An eigenanalysis together with the singular value decomposition is used to explain this phenomenon. The problem and its solution is a metaphor for matrix-based research in computational science and engineering.