From Random Polygon to Ellipse: An Eigenanalysis

Suppose x and y are unit 2-norm n-vectors whose components sum to zero. Let P(x, y) be the polygon obtained by connecting (x1, y1), . . . , (xn, yn), (x1, y1) in order. We say that P(x, y) is the normalized average of P(x,y) if it is obtained by connecting the midpoints of its edges and then normalizing the resulting vertex vectors x and y so that they have unit 2-norm. If this process is repeated starting with P0 = P(x(0), y(0)), then in the limit the vertices of the polygon iterates P(x(k), y(k)) converge to an ellipse 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 phenomena. The problem and its solution is a metaphor for matrix-based research in computational science and engineering. Key words. power method, eigenvalue analysis, ellipse, polygon AMS subject classifications. 15A18, 65F15

• Type: Journal
• Year: 2010
• Where: SIAMREV
• Authors: Adam N. Elmachtoub, Charles F. Van Loan
• Keywords: AMS Subject Classifications, Polygon, SIAMREV 2010, Unit 2-norm N-vectors