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?