Introduction

It is shown how historically the concept of pseudo-Hermiticity emerged from the studies of quantum mechanics Parity/Time reversal (PT) invariant systems. The properties of the metric that connects the operator to its adjoint in the pseudo-Hermitian condition are discussed. The random matrix theory (RMT) in its original form of Gaussian matrices introduced by E. Wigner whose classes are indexed by the values 1, 2, and 4 of the β Dyson index that fixes their symmetries is presented. The RMT tridiagonal matrices parametrized by a real positive value of β are introduced. The plan of the book is outlined.