Pseudo-Hermitian β-Hermite Ensemble with Real Eigenvalues

A short description of the Hermitian tridiagonal matrices of the β-Hermite ensemble with its main properties is given. It is then shown that the construction of the Hermitian ensemble can be performed by considering it as a problem of the theory of information using Maximum Entropy Principle to go from a full to a sparse matrix. As a bonus, this provides a framework to introduce the pseudo-Hermitian ensemble in which the Dyson index β plays a similar role it has in the Gaussian ensemble, namely, that of counting the number of independent off-diagonal elements.