1-Exp[β (Z[si,sj]-Z[r[si],sj])]

(∑_(r=-∞)^∞)Exp[-β((((si-sj))^2)-(((r-si-sj))^2))]

(Exp[-β((((r-s1-s2))^2)+(((s1-s2))^2))]-1)(Exp[-β((((r-s3-s4))^2)+(((s3-s4))^2))]-1)//Expand

Sum[(Exp[-β((((r-s1-s2))^2)+(((s1-s2))^2))]-1)/n,{r,-n,n},Assumptions->{s1∈Reals,s2∈Reals,s3∈Reals,s4∈Reals,β>0}]

m[{a_,b_,c_,d_,e_}]:={{a,b,c},{b,d,e},{c,e,-d-a}}

Total[((Flatten[m[{a,b,c,d,e}]])^2)]/.{a->(f Sqrt[1/3]+g),d->(f Sqrt[ 1/3]-g)}//FullSimplify

MatrixForm[test=#/Total[(((Flatten@#))^2)] &@m[RandomReal[{-1,1},5]]]

RotationMatrix[π,#].test.Inverse[RotationMatrix[π,#]]&@RandomReal[{-1,1},3]//MatrixForm

Det[test]

Total[Flatten[(({{a,b},{b,-a}})^2)]]

Total[((Flatten[Identity[2]-RotationMatrix[RandomReal[{0,2π}]].Inverse[RotationMatrix[RandomReal[{0,2π}]]]])^2)]

Plot[Total[((Flatten[Identity[2]-RotationMatrix[0].Inverse[RotationMatrix[θ]]])^2)],{θ,-π,π}]

Plot[Total[((Flatten[Identity[2]-RotationMatrix[0].Inverse[{{-1,0},{0,-1}}.RotationMatrix[θ]]])^2)],{θ,-π,π}]

Det[(({{a,b},{c,d}})^2)]

Plot[Det[(((RotationMatrix[0].Inverse[RotationMatrix[θ]]))^(3;))],{θ,-π,π}]

Plot[Total[((Flatten[Identity[2]-RotationMatrix[0].Inverse[{{-1,0},{0,-1}}.RotationMatrix[θ]]])^2)],{θ,-π,π}]

A=Array[a,{3,3}]

FullSimplify[Sqrt[x]Conjugate[Sqrt[x]]]

FullSimplify[(Tr[ConjugateTranspose[#].#])&@(A^(1-ϵ/2)),Assumptions->{ϵ∈Reals}]

Full

2

MatrixExp[A]

M=RandomVariate[CircularRealMatrixDistribution[12]];

Chop[ConjugateTranspose[#].#&@((M^(1-0.01)))]//MatrixForm

T[n_][θs_]:=Total[Map[ArrayPad[RotationMatrix[θs[[#+1]]],{{2#,n-2-2#},{2#,n-2-2#}}]&,Range[n/2]-1]]

θs=Map[θ[#]->RandomReal[WorkingPrecision->30]&,Range[6]-1];

P3=RandomVariate[CircularRealMatrixDistribution[12],WorkingPrecision->30];

M=Transpose[#].T[12].#&@P3;

Total[Flatten[((Abs[M])^(2-ϵ))]]/.θs/.ϵ->0.1

FullSimplify[Total[Flatten[(M^(2-ϵ))]],Assumptions->{ϵ>0}]

((Sum[p[i],{i,1,n}])^2)//Expand

Sum[Series[((Abs[x[i]])^(2-ϵ)),{ϵ,0,2}],{x,1,n}]//Expand

Limit[(x^2)((Log[Abs[x]])^n),x->0]

While[Det[M=RandomVariate[CircularRealMatrixDistribution[32]]]<0];

permutationMatrix[n_,pairs_]:=IdentityMatrix[n][[ReplacePart[Range[n],Flatten[{#2->#1}&@@@pairs]]]]

permuteMatrix[M_,pairs_]:=#.M.Transpose[#]&@permutationMatrix[Length[M],pairs]

moveEntry[M_,pair_]:=permuteMatrix[M,Join[{{pair[[2]],pair[[1]]+1}},{pair[[1]]+#,pair[[1]]+#+1}&/@Range[pair[[2]]-pair[[1]]-1]]]

permuteSchurDecomposition[P_,Q_]:={Fold[moveEntry,P,Partition[#,2]],Fold[moveEntry,Q,Partition[#,2]]}&@(First/@Position[Round[Abs[Diagonal[Q]],1 (10^(-6))],1])

getθs[Q_]:=ArcTan[Q[[1+2 #,1+2#]],Q[[2#+2,1+2#]]]&/@(Range[Length[Q]/2]-1)

root[M_]:=#1.T[Length[M]][getθs[#2]/2].Transpose[#1]&@@(permuteSchurDecomposition@@SchurDecomposition[M])

coursen[M_]:=#[[;;Length[M]/2,;;Length[M]/2]].#[[Length[M]/2+1;;,Length[M]/2+1;;]]&@root[M]

test=coursen[M];

test=M[[;;;;2,;;;;2]].M[[2;;;;2,2;;;;2]];

test.Transpose[test]//Chop