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Dan Piponi

The signed permutation matrices (SPMs) are permutation matrices except that you can use -1 instead of 1. Ie. they are matrices with just one non-zero entry in each row and each column and that entry is 1 or -1.

SPMs form a finite group so their eigenvalues must lie on the unit circle.

But what about the sum of two SPMs? Here's a plot of the eigenvalues of the sums of 2 8x8 SPMs.

(Sorry, top and bottom cropped, Mathematica has trouble plotting a million points even though seems like a pretty small number to me.)

What can I say? It's kind of like a disc filled with points with lots of smaller circles chopped out and an empty band across the equator and some wispy bits sticking out the sides.
Dan Piponi

A seasonal wreath using the eigenvalues of an SPM + 1/4 another.

Oct 01, 2023, 04:42 PM·Public·Web
2boosts·13favorites
Public
Dan Piponi

Code:

u[d_] := (2 RandomInteger[{0, 1}, d] - 1) IdentityMatrix[d][[
PermutationReplace[Range[d], RandomPermutation[d]]]]

z = Flatten[
Parallelize[
Table[Eigenvalues[u[8] + u[8]/4] // N, {i, 1, 500000}]]];

ListPlot[Transpose[{Re[z], Im[z]}], PlotStyle -> Opacity[0.0025],
AspectRatio -> 1]

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