I don't know what is meant by the normalisation constant of a matrix. But it looks from your suggested definitions 1] and 2] as though it is meant to be some function of the eigenvalues of a (positive?) matrix. The definition 1] gives the root mean square sum of the eigenvalues. The suggestion in 2] is to find the determinant (product of the eigenvalues).
It looks to me as though the formula for N given here ( ) is simply that N is the sum of the eigenvalues. This is equal to the trace of the matrix, which is the sum of its diagonal elements. That is easily seen to be . Or is that too simple-minded a suggestion?