Get positive eigenvalues of density matrix

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RM 2017-3-13

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评论： RM 2017-3-15

I would like to compute the eigenvector of the normalised Laplacian of a graph, which I think has properties of density matrix, that is trace = 1 and eigenvalues between 0 and 1. However when I compute the eigenvalues some are negative. Is there a problem with my code?

My initial adjacency matrix is:

A1 =  
   1     1     1     0     0     0     0
   0     1     1     0     0     0     0
   1     0     1     0     0     0     0
   1     1     0     0     0     0     0
   0     0     0     0     1     1     1
   0     0     0     1     0     1     1
   0     0     0     1     1     0     1
   0     0     0     1     1     1     0

My code then is:

[n,n] = size(A1);
D = zeros(n,1);
D(:,1) = sum(A1);
Norm1 = sum(D(:, 1));
L1 = (D(:, 1) - A1)/Norm1;
[V,D] = eig(L1)

The resulting eigenvalues are:

The first one is negative, but since L1 is a density matrix I think there is a problem with my code.

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David Goodmanson 2017-3-15

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Hi RM, I believe the problem is that the initial vector D is correct, but when it comes to defining the Laplacian matrix, D is supposed to be not a vector but a diagonal matrix with the elements of D on the diagonal, zero everywhere else. The diag command can create the one from the other. Defining L1 as

D = sum(A1);      % sum down columns of A1 to make a row vector
Dmat = diag(D);   % diagonal matrix
L1 = Dmat - A1;

does give a matrix whose eigenvalues are all positive or zero. Normalization is another question, since the Wikipedia definition, anyway, appears to differ from what you have.

Incidentally, the expression

D(:, 1) - A1

subtracts a matrix from a column vector. Before Matlab 2016b this would have given an error message. Now, because of the explicit expansion feature it does not (as long as the number of rows of D matches up with the number of rows in A1). It's a good feature but can lead to a whole new territory of unintended behavior.

RM 2017-3-15

Perfect! Thank you very much. This solves my problem.

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Answer 1

John D'Errico 2017-3-13

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编辑：John D'Errico 2017-3-13

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You cannot compute eigenvalues for a matrix subject to a constraint. I'm sorry, but that simply does not make mathematical sense.

In fact, even if I replace the elements of your matrix with 1/12, the eigenvalues still have one negative eigenvalue.

So lets see if it is just a precision thing? I've turned your matrix into a symbolic one, assuming that you really intended 1/12 when you show 0.0833.

Ms =
[  1/8, 1/12, 1/12, 1/12,  1/8,  1/8,  1/8,  1/8]
[ 1/12,  1/8, 1/12, 1/12,  1/8,  1/8,  1/8,  1/8]
[ 1/12, 1/12,  1/8, 1/12,  1/8,  1/8,  1/8,  1/8]
[ 1/12, 1/12, 1/12,  1/8,  1/8,  1/8,  1/8,  1/8]
[  1/8,  1/8,  1/8,  1/8,  1/8, 1/12, 1/12, 1/12]
[  1/8,  1/8,  1/8,  1/8, 1/12,  1/8, 1/12, 1/12]
[  1/8,  1/8,  1/8,  1/8, 1/12, 1/12,  1/8, 1/12]
[  1/8,  1/8,  1/8,  1/8, 1/12, 1/12, 1/12,  1/8]
eig(Ms)
ans =
 -1/8
 1/24
 1/24
 1/24
 1/24
 1/24
 1/24
  7/8

Lets see if this makes sense. I'll compute the eigenvector for that eigenvalue.

[V,D] = eig(M);
V(:,1)
ans =
      0.35355
      0.35355
      0.35355
      0.35355
     -0.35355
     -0.35355
     -0.35355
     -0.35355
M*V(:,1)
ans =
    -0.044194
    -0.044194
    -0.044194
    -0.044194
     0.044194
     0.044194
     0.044194
     0.044194

So, in fact, that negative eigenvalue makes complete sense. If I multiply your matrix by the vector V(:,1), then I get -1/8*V(:,1) back. That is what an eigenvalue/eigenvector pair means.

I would suggest you are in error when you claim that ALL of the eigenvalues of this matrix must be positive.

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RM 2017-3-14

Thank you for your answer. I wrote the question wrong I think. I may be computing the Laplacian of the adjacency matrix wrong and therefore having negative eigenvalues.