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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.)
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]
@dpiponi bottom is just a mirror of the top anyway it seems, yeah?
@dpiponi Somewhat reminiscent of what you get by looking at roots of polynomials with small-integer coefficients (see e.g. https://math.ucr.edu/home/baez/roots/).
(This isn't hugely surprising -- eigenvalues of matrices are roots of polynomials derived from their entries. I guess there are many ways of constraining polynomials' coefficients that lead to pictures that look a bit like these.)
@gjm It's a recurring theme on math social media :)
@gjm There are partial stories for the shapes of the details in these kinds of plots. Eg. for some of these kinds of plots you can approximate the computation with some simpler rational function of the matrix elements (which are by design, chosen to be integers). These approximate sets have to be subsets of Schmidt arrangements so you end up with regions that locally look a bit like that. https://math.katestange.net/illustration/schmidt-arrangements/