The Eigenvalues of a large matrix don't cross each other when plotted

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AVM 2020-9-25

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此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/599998-the-eigenvalues-of-a-large-matrix-don-t-cross-each-other-when-plotted

评论： Ameer Hamza 2020-9-26

I am trying to plot the eigenvalues of a matrix in accending order w.r.to some parameter. The matrix has large dimension. Here various eigenvalues of the matrix is not actually crossing at some points rather it seems to be repeling each other at those points. But it is expected that it should cross over each other at those points. I don't understand why this is happening. Please somebody help me. The matlab code is given below.

om=1.0;
om0=0;
dlt=0.5;
n=50;
I1=eye(n);
I2=eye(2);
matdimension= n-1;
tempvector = 0:1:matdimension;
tempvector = sqrt(tempvector);
tempmatrix = diag(tempvector);
anni= circshift(tempmatrix,-1);
crea = anni';
num=crea*anni;
c=crea+anni;
sigx=[0,1;1,0];
sigz=[1,0; 0,-1];
lm=0:0.01:1.0;
w1 = zeros(size(lm));
w2 = zeros(size(lm));
w3 = zeros(size(lm));
w4 = zeros(size(lm));
w5 = zeros(size(lm));
w6 = zeros(size(lm));
for i = 1:length(lm)   
      
       H= om*kron(I2,num) +(dlt./2)* kron(sigz,I1) + lm(i).*(kron(sigx,crea+anni));
   l= eig(H);
   v= sort(l);
    w1(i)=v(1); 
    w2(i)=v(2);
    w3(i)=v(3);
    w4(i)=v(4);
    w5(i)=v(5);
    w6(i)=v(6);
    
end
plot(lm,w1,'k',lm,w2,'r',lm,w3,'b',lm,w4,'g',lm,w5,'y',lm,w6,'c')

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Answer 1

Ameer Hamza 2020-9-25

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此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/599998-the-eigenvalues-of-a-large-matrix-don-t-cross-each-other-when-plotted#answer_500554

编辑：Ameer Hamza 2020-9-25

在 MATLAB Online 中打开

Yes, this is a very common problem with MATLAB's eig function, and John has created this excellent package to solve this problem: https://www.mathworks.com/matlabcentral/fileexchange/22885-eigenshuffle . This maintains consistency in the order of eigenvalues. Try the following code

om=1.0;
om0=0;
dlt=0.5;
n=50;
I1=eye(n);
I2=eye(2);
matdimension= n-1;
tempvector = 0:1:matdimension;
tempvector = sqrt(tempvector);
tempmatrix = diag(tempvector);
anni= circshift(tempmatrix,-1);
crea = anni';
num=crea*anni;
c=crea+anni;
sigx=[0,1;1,0];
sigz=[1,0; 0,-1];
lm=0:0.01:1.0;
H = zeros(100, 100, length(lm));
for i = 1:length(lm)      
       H(:,:,i) = om*kron(I2,num) +(dlt./2)* kron(sigz,I1) + lm(i).*(kron(sigx,crea+anni));
end
[~, e] = eigenshuffle(H);
e = flipud(e);
w1 = e(1,:);
w2 = e(2,:);
w3 = e(3,:);
w4 = e(4,:);
w5 = e(5,:);
w6 = e(6,:);
plot(lm,w1,'k',lm,w2,'r',lm,w3,'b',lm,w4,'g',lm,w5,'y',lm,w6,'c')