Eigenvalue

Author

Eric W. Weisstein
November 27, 2002

This notebook downloaded from http://mathworld.wolfram.com/notebooks/LinearAlgebra/Eigenvalue.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/Eigenvalue.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Analytic

2x2

A=Array[(Subscript[a, #1,#2])&,{2,2}]

TraditionalForm[x1=Collect[CharacteristicPolynomial[A,x],x]]

Solve[x1==0,x]//FullSimplify

Eigenvalues[A]//FullSimplify

x2=Collect[Simplify[x^2-Tr[A]x+Det[A]],x]

Simplify[x1-x2]

3x3

A=Array[(Subscript[a, #1,#2])&,{3,3}]

x1=-Collect[CharacteristicPolynomial[A,x],x]

x2=Collect[Simplify[x^3-Tr[A]x^2-Sum[((Subscript[a, i,j])(Subscript[a, j,i])-(Subscript[a, i,i])(Subscript[a, j,j]))(1-KroneckerDelta[i,j]),{i,3},{j,3}]x/2-Det[A]],x]

Simplify[x1-x2]

Eigenvalue Distributions

<<Graphics`Graphics`;
<<Statistics`DataManipulation`;

barLabels[x0_,x1_,{dx_,ddx_}]:=Module[{i},
Flatten[Table[{ToString[i],Table["",{dx/ddx-1}]},{i,x0,x1,dx}]]
]

SquareDistribution[num_]:=Table[Random[Real,{-1,1}],{num},{2}]

CircularDistribution[num_]:=Table[Sqrt[Random[]]{Cos[#],Sin[#]}&//@Random[Real,{0,2Pi}],{num}]

RadialDistribution[num_]:=Table[Random[]{Cos[#],Sin[#]}&//@Random[Real,{0,2Pi}],{num}]

SphericalNonUniformDistribution[num_]:=Table[{Cos[#1]Sin[#2],Sin[#1]Sin[#2],Cos[#2]}&[Random[Real,{0,2Pi}],Random[Real,{0,Pi}]],{num}]

SphericalDistribution[num_]:=Table[{Cos[#1]Sqrt[1-#2^2],Sin[#1]Sqrt[1-#2^2],#2}&[Random[Real,{0,2Pi}],Random[Real,{-1,1}]],{num}]

CircularEigenvalues[n_,num_]:=Eigenvalues/@Partition[Partition[CircularDistribution[n^2 num].{1,I},n],n]

SquareEigenvalues[n_,num_]:=Eigenvalues/@Partition[Partition[SquareDistribution[n^2 num].{1,I},n],n]

2x2

CharacteristicPolynomial[{{(Subscript[a, 11]),(Subscript[a, 12])},{(Subscript[a, 21]),(Subscript[a, 22])}},x]

CoefficientList[CharacteristicPolynomial[{{(Subscript[a, 11]),(Subscript[a, 12])},{(Subscript[a, 21]),(Subscript[a, 22])}},x],x]

Det[{{(Subscript[a, 11]),(Subscript[a, 12])},{(Subscript[a, 21]),(Subscript[a, 22])}}]

Tr[{{(Subscript[a, 11]),(Subscript[a, 12])},{(Subscript[a, 21]),(Subscript[a, 22])}}]

FullSimplify[Eigenvalues[{{(Subscript[a, 11]),(Subscript[a, 12])},{(Subscript[a, 21]),(Subscript[a, 22])}}]]

Eigenvalues[{{(Subscript[a, 11]),(Subscript[a, 12])},{(Subscript[a, 21]),(Subscript[a, 22])}}]/.{(Subscript[a, 11])->(Subscript[a, 111])+I (Subscript[a, 112]),(Subscript[a, 12])->(Subscript[a, 121])+I (Subscript[a, 122]),(Subscript[a, 21])->(Subscript[a, 211])+I (Subscript[a, 212]),(Subscript[a, 22])->(Subscript[a, 221])+I (Subscript[a, 222])}

FullSimplify[ComplexExpand[Abs[%47],TargetFunctions->{Re,Im}]]

3x3

Collect[CharacteristicPolynomial[{{(Subscript[a, 11]),(Subscript[a, 12]),(Subscript[a, 13])},{(Subscript[a, 21]),(Subscript[a, 22]),(Subscript[a, 23])},{(Subscript[a, 31]),(Subscript[a, 32]),(Subscript[a, 33])}},x],x]

Det[{{(Subscript[a, 11]),(Subscript[a, 12]),(Subscript[a, 13])},{(Subscript[a, 21]),(Subscript[a, 22]),(Subscript[a, 23])},{(Subscript[a, 31]),(Subscript[a, 32]),(Subscript[a, 33])}}]

Det[{{(Subscript[a, 11]),(Subscript[a, 12]),(Subscript[a, 13])},{(Subscript[a, 21]),(Subscript[a, 22]),(Subscript[a, 23])},{(Subscript[a, 31]),(Subscript[a, 32]),(Subscript[a, 33])}}]

Sum[(a[i,j]a[j,i]-a[i,i]a[j,j])(1-KroneckerDelta[i,j])/2,{i,3},{j,3}]

FullSimplify[Eigenvalues[{{(Subscript[a, 11]),(Subscript[a, 12]),(Subscript[a, 13])},{(Subscript[a, 21]),(Subscript[a, 22]),(Subscript[a, 23])},{(Subscript[a, 31]),(Subscript[a, 32]),(Subscript[a, 33])}}]]

CoefficientList[CharacteristicPolynomial[{{(Subscript[a, 11]),(Subscript[a, 12]),(Subscript[a, 13])},{(Subscript[a, 21]),(Subscript[a, 22]),(Subscript[a, 23])},{(Subscript[a, 31]),(Subscript[a, 32]),(Subscript[a, 33])}},x],x]

Square Distribution

Max[Abs[#]&/@Flatten[SquareEigenvalues[1,1000000]]]

Max[Abs[#]&/@Flatten[SquareEigenvalues[2,500000]]]

Max[Abs[#]&/@Flatten[SquareEigenvalues[3,200000]]]

Max[Abs[#]&/@Flatten[SquareEigenvalues[4,200000]]]

Max[Abs[#]&/@Flatten[SquareEigenvalues[5,50000]]]

Max[Abs[#]&/@Flatten[SquareEigenvalues[6,50000]]]

Max[Abs[#]&/@Flatten[SquareEigenvalues[7,50000]]]

Max[Abs[#]&/@Flatten[SquareEigenvalues[8,20000]]]

l={1.41369,2.531,3.13637,3.36742,3.59846,3.7985,3.96897,4.0755};

ListPlot[l,PlotJoined->True]

Plot

Circular Distribution

Max[Abs[#]&/@Flatten[CircularEigenvalues[1,100000]]]

2x2

d2=Abs[#]&/@Flatten[CircularEigenvalues[2,50000]];

Max[d2]

chart2=BarChart[BinCounts[d2,{0,2,.05}],PlotRange->All,BarSpacing->0,
BarLabels->barLabels[0,2,{.25,.05}]]

3x3

d3=Abs[#]&/@Flatten[CircularEigenvalues[3,50000]];

Max[d3]

chart3=BarChart[BinCounts[d3,{0,2.5,.05}],PlotRange->All,BarSpacing->0,BarLabels->barLabels[0,2.5,{.5,.05}]]

4x4

d4=Abs[#]&/@Flatten[CircularEigenvalues[4,50000]];

Max[d4]

chart4=BarChart[BinCounts[d4,{0,2.5,.05}],PlotRange->All,BarSpacing->0,BarLabels->barLabels[0,2.8,{.5,.05}]]

Show[GraphicsArray[{chart2,chart3,chart4}]]

5x5 and up

Max[Abs[#]&/@Flatten[CircularEigenvalues[5,50000]]]

Max[Abs[#]&/@Flatten[CircularEigenvalues[6,50000]]]

Max[Abs[#]&/@Flatten[CircularEigenvalues[7,50000]]]

lc={1,1.92391,2.48778,2.79541,2.97896,3.1623,3.3119};

Show[Plot[1+Sqrt[x-1],{x,1,7}],
ListPlot[lc,PlotJoined->True]
]

Fit[lc,{1,Log[x]},x]

ListPlot[lc^2,PlotJoined->True]