File:Random matrix eigenvalues.gif

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Original file ‎(800 × 800 pixels, file size: 4.7 MB, MIME type: image/gif, looped, 100 frames, 10 s)

Captions

Captions

Eigenvalues of 4 classes of random matrices on the complex plane.

Summary[edit]

Description
English: Eigenvalues of 4 classes of random matrices on the complex plane. (I am using ~10³ 2x2 matrices, so any central limit theorem valid for large matrices do not really apply here)
Date
Source https://twitter.com/j_bertolotti/status/1294255802296016898
Author Jacopo Bertolotti
Permission
(Reusing this file)		 https://twitter.com/j_bertolotti/status/1030470604418428929

Mathematica 12.1 code[edit]

\[Lambda]random = {};
Do[m = 1/2 (RandomVariate[NormalDistribution[0, 1], {2, 2}] + I RandomVariate[NormalDistribution[0, 1], {2, 2}]);
  AppendTo[\[Lambda]random, Eigenvalues[m]]; , {1000}];
\[Lambda]ortho = {};
Do[m = 1/2 (RandomVariate[NormalDistribution[0, 1], {2, 2}] + I RandomVariate[NormalDistribution[0, 1], {2, 2}]);
  m = Orthogonalize[m]; AppendTo[\[Lambda]ortho, Eigenvalues[m]]; , {1000}];
\[Lambda]symp = {};
Do[m = 1/2 (RandomVariate[NormalDistribution[0, 1], {2, 2}] + I RandomVariate[NormalDistribution[0, 1], {2, 2}]);
  m = m/Sqrt[Det[m]]; AppendTo[\[Lambda]symp, Eigenvalues[m]]; , {1000}];
\[Lambda]herm = {};
Do[m = 1/2 (RandomVariate[NormalDistribution[0, 1], {2, 2}] + I RandomVariate[NormalDistribution[0, 1], {2, 2}]);
  m = (m + ConjugateTranspose[m])/2; AppendTo[\[Lambda]herm, Eigenvalues[m]]; , {1000}];
step[t_] := Ceiling[1000*t^2]
p0 = Table[
   GraphicsGrid[{{
      ListPlot[
       Evaluate[{Re[#], Im[#]} & /@ Flatten[\[Lambda]random[[1 ;; step[t]]] ] ], PlotRange -> {{-3, 3}, {-3, 3}}, AspectRatio -> 1, AxesStyle -> {Black, Thick}, PlotMarkers -> {Graphics[{Orange, Disk[], Black, Circle[]}], 0.03 }, AxesLabel -> {"Re[\[Lambda]]", "Im[\[Lambda]]"}, LabelStyle -> {Black, Bold}, AxesStyle -> Black, PlotLabel -> "Gaussian noise"]
      ,
      ListPlot[
       Evaluate[{Re[#], Im[#]} & /@ Flatten[\[Lambda]ortho[[1 ;; step[t]]]] ], PlotRange -> {{-1.3, 1.3}, {-1.3, 1.3}}, AspectRatio -> 1, AxesStyle -> {Black, Thick}, PlotMarkers -> {Graphics[{Orange, Disk[], Black, Circle[]}], 0.03 }, AxesLabel -> {"Re[\[Lambda]]", "Im[\[Lambda]]"}, LabelStyle -> {Black, Bold}, AxesStyle -> Black, PlotLabel -> "Orthogonal/Unitary"]
      }, {
      ListPlot[
       Evaluate[{Re[#], Im[#]} & /@ Flatten[\[Lambda]herm[[1 ;; step[t]]]] ], PlotRange -> {{-3, 3}, {-3, 3}}, AspectRatio -> 1, AxesStyle -> {Black, Thick}, PlotMarkers -> {Graphics[{Orange, Disk[], Black, Circle[]}], 0.03 }, AxesLabel -> {"Re[\[Lambda]]", "Im[\[Lambda]]"}, LabelStyle -> {Black, Bold}, AxesStyle -> Black, PlotLabel -> "Symmetric/Hermitian"]
      ,
      ListPlot[
       Evaluate[{Re[#], Im[#]} & /@ Flatten[\[Lambda]symp[[1 ;; step[t]]]] ], PlotRange -> {{-3, 3}, {-3, 3}}, AspectRatio -> 1, AxesStyle -> {Black, Thick}, PlotMarkers -> {Graphics[{Orange, Disk[], Black, Circle[]}], 0.03 }, AxesLabel -> {"Re[\[Lambda]]", "Im[\[Lambda]]"}, LabelStyle -> {Black, Bold}, AxesStyle -> Black, PlotLabel -> "Symplectic"]
      }}, ImageSize -> 800]
   , {t, 0.0001, 1, 0.01}];
ListAnimate[p0]

Licensing[edit]

I, the copyright holder of this work, hereby publish it under the following license:
Creative Commons CC-Zero This file is made available under the Creative Commons CC0 1.0 Universal Public Domain Dedication.
The person who associated a work with this deed has dedicated the work to the public domain by waiving all of their rights to the work worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law. You can copy, modify, distribute and perform the work, even for commercial purposes, all without asking permission.

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Date/TimeThumbnailDimensionsUserComment
current09:06, 17 August 2020Thumbnail for version as of 09:06, 17 August 2020800 × 800 (4.7 MB)Berto (talk | contribs)Uploaded own work with UploadWizard

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