1720.sagews

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 $... a_{-n}, ..., a_0, a_1, ...$.  We will only use $2n+1$ elements, $n$ on each side of $a_0$.

## make the underlying sequence a_{-n} .... a_0 ... a_n
n=20 # size of sequence

seq = []

## sample sequence is simple 1/k
for k in xrange(-n,n+1):
    if k==0: seq.append(0)
    else: seq.append(1/k)

print "seq is ", seq

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 $n$.

## build a list that will be converted to a matrix
B = []
j=n
for i in xrange(0,n+1):
    B.append(seq[j:j+n+1])  # slice through the sequence to get rows
    j=j-1

show(B)

\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.  

A=matrix(B) 
show(A)

\newcommand{\Bold}[1]{\mathbf{#1}}\left($$\begin{array}{rrrrrrrrrrrrrrrrrrrrr} 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} \\ -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} \\ -\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} \\ -\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} \\ -\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} \\ -\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} \\ -\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} \\ -\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} \\ -\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} \\ -\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} \\ -\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} \\ -\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} \\ -\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} \\ -\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} \\ -\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} \\ -\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} \\ -\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} \\ -\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} \\ -\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} \\ -\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 \\ -\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 \end{array}$$\right)

Among other things, this means I can get easily get eigenvalues.

E=A.eigenvalues()
E

[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.

# convert complex numbers to (x,y) for graphing.
def c2r(z):
    return (real(z),imaginary(z))

L1=[]  # make a list of (x,y) points for plotting 
for i in E:
    L1.append(c2r(i))


G=list_plot(L1, color="red")
show(G)