Visualize repelling behavior of eigenvalues

Eigenvalues of matrices don’t like to be next to each other. Similar behavior underlies behavior of primes, energy levels and tree rings, see https://www.americanscientist.org/article/the-spectrum-of-riemannium

Below is a simple visualization of repelling behavior -- take a random matrix, and visualize what happens with eigenvalues as we rotate this matrix.

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d=20;SeedRandom[1];mat=RandomVariate[NormalDistribution[],{d,d}]

d

+.1IdentityMatrix[d]I;rotate[theta_]:=(blocks=Table[R[i]->RotationMatrix[theta],{i,d/2}];mat.ArrayFlatten[DiagonalMatrix[Array[R,d/2]]/.blocks]);Animate[ComplexListPlot[Eigenvalues[rotate[p]],Axes->None,PlotRange->{-1-I,1+I}],{p,0,2Pi},SaveDefinitions->True]

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p

We can visualize the trajectories

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d=20;evals=Table[Eigenvalues[rotate[p]],{p,0,2Pi,.001}];ListPlot[ReIm@Flatten@evals,PlotStyle->Directive[PointSize[.01],Opacity[.01]],PlotRange->{{-1,1},{-1,1}},AspectRatio->1,Axes->None]

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