In[]:=
CompoundExpression[
]
​​deploy
Sat 4 Jan 2025 15:59:40
Verify solution of Xiangxiang: https://math.stackexchange.com/a/5017119/998​
Use P1 and P2 formulas
​
Parent: linear-estimation-scratch-2025.nb/Chris DeSa xiangxiang verify 04jan25
In[]:=
Clear["Global`*"];​​Needs["MaTeX`"];​​​​numSamples=100000;​​​​​​trs[A_,s_Integer]:=With[{As=MatrixPower[A,s]},Tr[As]];​​norms[A_,s_Integer]:=With[{As=MatrixPower[A,s]},
2
Norm[As,"Frobenius"]
];​​sample[d_]:=Module[{A},​​A=N@IdentityMatrix[d];​​Do[LinearAlgebra`LAPACK`LARF["R",v,1.,A],{v,Normalize/@RandomVariate[NormalDistribution[],{d,d}]}];​​A​​];​​​​Needs["Notation`"];​​Notation
s_
A_
⟺
MatrixPower[A_,s_]
​​​​s0=2;​​d0=3;​​SF=StringForm;​​​​Print[SF["s=``, d=``, numSamples=``",s0,d0,numSamples]];​​​​​​eval[func_]:=(N@Mean@Table[func/.{A->sample[d0],d->d0,s->s0},{numSamples}]);​​formula1=(d^2+d-2)/2*(1-2/d+2/(d*(d+2)))^d-(d^2-d)/2*(1-2/d)^d+(1-1/d)^d;​​formula2=(d^2+2d)/3*(1-2/d+3/(d*(d+2)))^d-(d^2-d)/3*(1-2/d)^d;​​formulas={Tr[
s
A
],Tr[
s
(A.A)
],formula1,formula2};​​​​Unprotect[MaTeX];​​MaTeX[Tr[
s
A
]]=MaTeX["\\operatorname{Tr} A^s "];​​MaTeX[Tr[
s
(A.A)
]]=MaTeX["\\operatorname{Tr}(AA^T)^s"];​​​​vals=Table[eval/@formulas,{10}];​​TableForm[vals,TableHeadings->{{},(MaTeX/@formulas[[;;2]])~Join~{"formulaP1","formulaP2"}}]​​GraphicsRow@MaTeX[{"\\text{formulaP1 =}",formula1}]​​GraphicsRow@MaTeX[{"\\text{formulaP2 =}",formula2}]​​
s=2​, d=3​, numSamples=100000
Out[]//TableForm=
formulaP1
formulaP2
0.692547
0.683769
0.693333
0.684444
0.692136
0.684885
0.693333
0.684444
0.69336
0.686362
0.693333
0.684444
0.694073
0.684083
0.693333
0.684444
0.694847
0.685155
0.693333
0.684444
0.695242
0.683373
0.693333
0.684444
0.69355
0.683157
0.693333
0.684444
0.691748
0.684751
0.693333
0.684444
0.694269
0.684857
0.693333
0.684444
0.693769
0.684539
0.693333
0.684444
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Out[]=