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A question about a shuffled vs. unshuffled correlation matrix

I took the correlation matrix and shuffled between its values symmetrically. (shuffled only the left lower triangle of the matrix and copied the values to the upper right triangle. saving the diagonal of ones untouched) The eigenvalues spectrum of both the correlation matrix and the shuffled matrix have one large eigenvalue. As I understand from my results, this eigenvalue reflect a uniform collective mode of the fluctuations (because most correlation values are positive and are the same for the shuffled matrix)

However, the eigenvector of these first eigenvalues is different: For the correlation matrix, most are positive and very distributed. For the shuffled matrix, all of them are positive and are centered around one positive value. (Of course that if by chance the shuffling was the same as the correlation matrix I would have reached similar results...

Can somebody please explain to me this phenomenon?

EDIT: Let's say that I have a correlation matrix between the neural response of 10 neurons -> a 10X10 correlation matrix with a diagonal of ones. I extract all elements from the matrix (I have 45 elements which appear twice Cij=Cji). Then I randomly insert these elements into a new matrix while saving the diagonal of ones and the symatry of the matrix: Cij=Cji. Is that clear now?

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    $\begingroup$ Could you please be more specific about your "shuffles"? The only ones that are meaningful for correlation matrices simultaneously permute rows and columns, but you haven't really mentioned the columns. That is, for a symmetric $n$ by $n$ matrix $(a_{i,j})$ and a permutation $\sigma$, the correctly "shuffled" matrix is $(a_{\sigma(i),\sigma(j)})$, whereas your method seems to indicate you have obtained $(a_{\sigma(i),j})$ for $i\ge j$ and $(a_{i,\sigma(j)})$ otherwise. $\endgroup$ – whuber Mar 5 '13 at 16:02
  • $\begingroup$ Why are you doing this? It cannot be to simulate a distribution of correlation matrices, because you have a good chance of producing matrices that are not correlation matrices (they will be indefinite). In fact, if the matrix is sufficiently large, the vast majority of shuffled versions of it will not be valid correlation matrices. $\endgroup$ – whuber Mar 6 '13 at 21:25
  • $\begingroup$ I'm doing it in order to compare my correlation matrix to a different matrix which has identical values, but it's not a correlation matrix. this way I can better examine the correlation structure. Regarding eigenvalues, I assume that this one large eigenvalue in both the correlation matrix and the shuffled matrix is an outcome of the uniform correlation I have (almost all of my correlations are positive), but what I dont understand is why the elements of the eigenvector corresponding to the largest eigenvalue is so different between the two matrices: $\endgroup$ – Omri374 Mar 10 '13 at 6:40
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    $\begingroup$ Thanks for explaining. I am at a loss, though, to understand how this process of permuting the matrix elements reveals anything at all about "correlation structure." About the only properties of the original matrix left unchanged are (1) its symmetry and (2) its $L^p$ norms. The latter might be at the root of the behavior you observe concerning the eigenvalues. $\endgroup$ – whuber Mar 10 '13 at 14:47

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