Inward matrix products, generalised density functions and Rayleigh–Schrödinger perturbation theory

Abstract

A matrix product, the inward product, of two matrices is defined as an operation of internal composition involving two (m×n)-dimensional matrices and yielding another matrix of the same dimension. Such a product, known as Hadamard or Schur product in literature, presents typical properties and corresponds to a usual matrix product, within the isomorphic set of (mn)-dimensional diagonal matrices. It can be directly used to construct generalised density functions. A useful application to Rayleigh–Schrödinger perturbation theory is also discussed.