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The statement about general matrix diagonalisation is not true (not every matrix is similiar to a diagonal matrix, take nilpotent matrix as counter example). F
by andi999 6y ago
The statement about general matrix diagonalisation is not true (not every matrix is similiar to a diagonal matrix, take nilpotent matrix as counter example).
For random matrice the author introduces what it means to be rotationally (O) invariant, but is what follow restricted to that? Also it is not too clear why the code is actually doing (creating the same ensemble) as what was talked about before.
- bline80 6y agoHi Andi, thanks for your comment, I just found out about this. Actually I did not write that any matrix can be diagonalised. I wrote that we are dealing with matrices that can be diagonalised via an orthogonal transformation. Indeed, I also specify (in the notebook is quite clear) that we are working with real symmetric matrices, and these can be diagonalised via orthogonal transformations: https://en.wikipedia.org/wiki/Diagonalizable_matrix#Diagonalizable_matrices https://en.wikipedia.org/wiki/Diagonalizable_matrix#Diagonal... This is why is called the Gaussian Orthogonal Ensemble. The Unitary ensemble deals with unitary matrices for example. All of these ensembles deal with diagonalisable matrices. The code is providing a numerical verification of the analytical formula. It is a simple code that samples symmetric matrices of different NxN size from a Gaussian distribution. For N fixed, you draw a certain number of samples and you plot the eigenvalue distribution and compare it with the analytics. Then you can see that for small N there is a deviation from the analytical expression, as this was formally obtained in the large N limit. Hope this clarifies.
- andi999 6y agoThanks. By saying 'NxN matrices from a Gaussian distribution' you mean each entry of the matrix is a sample of a Gaussian distribution, right? At least this is what the code is doing(there could be other meanings to this term)