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At the Golden Section MAA meeting I heard a nice talk by Estelle Basor. I saw some nice pictures and decided to try making some with Sage.
A Toepliz Matrix starts with some sequence . We will only use elements, on each side of .
seq is [-1/20, -1/19, -1/18, -1/17, -1/16, -1/15, -1/14, -1/13, -1/12, -1/11, -1/10, -1/9, -1/8, -1/7, -1/6, -1/5, -1/4, -1/3, -1/2, -1, 0, 1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9, 1/10, 1/11, 1/12, 1/13, 1/14, 1/15, 1/16, 1/17, 1/18, 1/19, 1/20]
Take our sequence and build a Python array (a list of lists). Python lets me "slice" through my sequence with a window size of .
\newcommand{\Bold}[1]{\mathbf{#1}}\left[\left[0, 1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \frac{1}{6}, \frac{1}{7}, \frac{1}{8}, \frac{1}{9}, \frac{1}{10}, \frac{1}{11}, \frac{1}{12}, \frac{1}{13}, \frac{1}{14}, \frac{1}{15}, \frac{1}{16}, \frac{1}{17}, \frac{1}{18}, \frac{1}{19}, \frac{1}{20}\right], \left[-1, 0, 1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \frac{1}{6}, \frac{1}{7}, \frac{1}{8}, \frac{1}{9}, \frac{1}{10}, \frac{1}{11}, \frac{1}{12}, \frac{1}{13}, \frac{1}{14}, \frac{1}{15}, \frac{1}{16}, \frac{1}{17}, \frac{1}{18}, \frac{1}{19}\right], \left[-\frac{1}{2}, -1, 0, 1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \frac{1}{6}, \frac{1}{7}, \frac{1}{8}, \frac{1}{9}, \frac{1}{10}, \frac{1}{11}, \frac{1}{12}, \frac{1}{13}, \frac{1}{14}, \frac{1}{15}, \frac{1}{16}, \frac{1}{17}, \frac{1}{18}\right], \left[-\frac{1}{3}, -\frac{1}{2}, -1, 0, 1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \frac{1}{6}, \frac{1}{7}, \frac{1}{8}, \frac{1}{9}, \frac{1}{10}, \frac{1}{11}, \frac{1}{12}, \frac{1}{13}, \frac{1}{14}, \frac{1}{15}, \frac{1}{16}, \frac{1}{17}\right], \left[-\frac{1}{4}, -\frac{1}{3}, -\frac{1}{2}, -1, 0, 1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \frac{1}{6}, \frac{1}{7}, \frac{1}{8}, \frac{1}{9}, \frac{1}{10}, \frac{1}{11}, \frac{1}{12}, \frac{1}{13}, \frac{1}{14}, \frac{1}{15}, \frac{1}{16}\right], \left[-\frac{1}{5}, -\frac{1}{4}, -\frac{1}{3}, -\frac{1}{2}, -1, 0, 1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \frac{1}{6}, \frac{1}{7}, \frac{1}{8}, \frac{1}{9}, \frac{1}{10}, \frac{1}{11}, \frac{1}{12}, \frac{1}{13}, \frac{1}{14}, \frac{1}{15}\right], \left[-\frac{1}{6}, -\frac{1}{5}, -\frac{1}{4}, -\frac{1}{3}, -\frac{1}{2}, -1, 0, 1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \frac{1}{6}, \frac{1}{7}, \frac{1}{8}, \frac{1}{9}, \frac{1}{10}, \frac{1}{11}, \frac{1}{12}, \frac{1}{13}, \frac{1}{14}\right], \left[-\frac{1}{7}, -\frac{1}{6}, -\frac{1}{5}, -\frac{1}{4}, -\frac{1}{3}, -\frac{1}{2}, -1, 0, 1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \frac{1}{6}, \frac{1}{7}, \frac{1}{8}, \frac{1}{9}, \frac{1}{10}, \frac{1}{11}, \frac{1}{12}, \frac{1}{13}\right], \left[-\frac{1}{8}, -\frac{1}{7}, -\frac{1}{6}, -\frac{1}{5}, -\frac{1}{4}, -\frac{1}{3}, -\frac{1}{2}, -1, 0, 1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \frac{1}{6}, \frac{1}{7}, \frac{1}{8}, \frac{1}{9}, \frac{1}{10}, \frac{1}{11}, \frac{1}{12}\right], \left[-\frac{1}{9}, -\frac{1}{8}, -\frac{1}{7}, -\frac{1}{6}, -\frac{1}{5}, -\frac{1}{4}, -\frac{1}{3}, -\frac{1}{2}, -1, 0, 1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \frac{1}{6}, \frac{1}{7}, \frac{1}{8}, \frac{1}{9}, \frac{1}{10}, \frac{1}{11}\right], \left[-\frac{1}{10}, -\frac{1}{9}, -\frac{1}{8}, -\frac{1}{7}, -\frac{1}{6}, -\frac{1}{5}, -\frac{1}{4}, -\frac{1}{3}, -\frac{1}{2}, -1, 0, 1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \frac{1}{6}, \frac{1}{7}, \frac{1}{8}, \frac{1}{9}, \frac{1}{10}\right], \left[-\frac{1}{11}, -\frac{1}{10}, -\frac{1}{9}, -\frac{1}{8}, -\frac{1}{7}, -\frac{1}{6}, -\frac{1}{5}, -\frac{1}{4}, -\frac{1}{3}, -\frac{1}{2}, -1, 0, 1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \frac{1}{6}, \frac{1}{7}, \frac{1}{8}, \frac{1}{9}\right], \left[-\frac{1}{12}, -\frac{1}{11}, -\frac{1}{10}, -\frac{1}{9}, -\frac{1}{8}, -\frac{1}{7}, -\frac{1}{6}, -\frac{1}{5}, -\frac{1}{4}, -\frac{1}{3}, -\frac{1}{2}, -1, 0, 1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \frac{1}{6}, \frac{1}{7}, \frac{1}{8}\right], \left[-\frac{1}{13}, -\frac{1}{12}, -\frac{1}{11}, -\frac{1}{10}, -\frac{1}{9}, -\frac{1}{8}, -\frac{1}{7}, -\frac{1}{6}, -\frac{1}{5}, -\frac{1}{4}, -\frac{1}{3}, -\frac{1}{2}, -1, 0, 1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \frac{1}{6}, \frac{1}{7}\right], \left[-\frac{1}{14}, -\frac{1}{13}, -\frac{1}{12}, -\frac{1}{11}, -\frac{1}{10}, -\frac{1}{9}, -\frac{1}{8}, -\frac{1}{7}, -\frac{1}{6}, -\frac{1}{5}, -\frac{1}{4}, -\frac{1}{3}, -\frac{1}{2}, -1, 0, 1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \frac{1}{6}\right], \left[-\frac{1}{15}, -\frac{1}{14}, -\frac{1}{13}, -\frac{1}{12}, -\frac{1}{11}, -\frac{1}{10}, -\frac{1}{9}, -\frac{1}{8}, -\frac{1}{7}, -\frac{1}{6}, -\frac{1}{5}, -\frac{1}{4}, -\frac{1}{3}, -\frac{1}{2}, -1, 0, 1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}\right], \left[-\frac{1}{16}, -\frac{1}{15}, -\frac{1}{14}, -\frac{1}{13}, -\frac{1}{12}, -\frac{1}{11}, -\frac{1}{10}, -\frac{1}{9}, -\frac{1}{8}, -\frac{1}{7}, -\frac{1}{6}, -\frac{1}{5}, -\frac{1}{4}, -\frac{1}{3}, -\frac{1}{2}, -1, 0, 1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}\right], \left[-\frac{1}{17}, -\frac{1}{16}, -\frac{1}{15}, -\frac{1}{14}, -\frac{1}{13}, -\frac{1}{12}, -\frac{1}{11}, -\frac{1}{10}, -\frac{1}{9}, -\frac{1}{8}, -\frac{1}{7}, -\frac{1}{6}, -\frac{1}{5}, -\frac{1}{4}, -\frac{1}{3}, -\frac{1}{2}, -1, 0, 1, \frac{1}{2}, \frac{1}{3}\right], \left[-\frac{1}{18}, -\frac{1}{17}, -\frac{1}{16}, -\frac{1}{15}, -\frac{1}{14}, -\frac{1}{13}, -\frac{1}{12}, -\frac{1}{11}, -\frac{1}{10}, -\frac{1}{9}, -\frac{1}{8}, -\frac{1}{7}, -\frac{1}{6}, -\frac{1}{5}, -\frac{1}{4}, -\frac{1}{3}, -\frac{1}{2}, -1, 0, 1, \frac{1}{2}\right], \left[-\frac{1}{19}, -\frac{1}{18}, -\frac{1}{17}, -\frac{1}{16}, -\frac{1}{15}, -\frac{1}{14}, -\frac{1}{13}, -\frac{1}{12}, -\frac{1}{11}, -\frac{1}{10}, -\frac{1}{9}, -\frac{1}{8}, -\frac{1}{7}, -\frac{1}{6}, -\frac{1}{5}, -\frac{1}{4}, -\frac{1}{3}, -\frac{1}{2}, -1, 0, 1\right], \left[-\frac{1}{20}, -\frac{1}{19}, -\frac{1}{18}, -\frac{1}{17}, -\frac{1}{16}, -\frac{1}{15}, -\frac{1}{14}, -\frac{1}{13}, -\frac{1}{12}, -\frac{1}{11}, -\frac{1}{10}, -\frac{1}{9}, -\frac{1}{8}, -\frac{1}{7}, -\frac{1}{6}, -\frac{1}{5}, -\frac{1}{4}, -\frac{1}{3}, -\frac{1}{2}, -1, 0\right]\right]
Now we go from Python to Sage. My Python array becomes a Sage matrix.
\newcommand{\Bold}[1]{\mathbf{#1}}\left(\right)
Among other things, this means I can get easily get eigenvalues.
[0, -2.788123013295495?*I, -2.50741672529829?*I, -2.22932873886824?*I, -1.95144660499993?*I, -1.673337201058448?*I, -1.394935291487221?*I, -1.116267748108292?*I, -0.8373854786662517?*I, -0.5583439717908066?*I, -0.2791978568704422?*I, 0.2791978568704422?*I, 0.5583439717908066?*I, 0.8373854786662517?*I, 1.116267748108292?*I, 1.394935291487221?*I, 1.673337201058448?*I, 1.95144660499993?*I, 2.22932873886824?*I, 2.50741672529829?*I, 2.788123013295495?*I]
Notice that the eigenvalues are complex numbers. In fact, they all live on the line real=0.