Hi 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#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.