File:Random matrix eigenvalues.gif
Summary
| 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
\[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
I, the copyright holder of this work, hereby publish it under the following license:
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