Inward matrix products:: extensions and applications to quantum mechanical foundations of QSAR

A special matrix product, the inward product, of two matrices is defined as an operation of internal composition in matrix spaces. A generalisation and an extension of the inward product make it a very flexible algorithmic tool. This product, known from a long time, also called Hadamard or Schur product, presents characteristic properties, which made such an operation interesting enough due to its potential use in many fields, such as quantum chemistry, where matrix manipulation is a common trait. Here, besides the introduction to the main features of inward product, it is shown that there is a wide prospect of applications, as well as evident connections with other mathematical structures like tagged sets and vector semispaces, useful in theoretical chemistry applications. Approximate least-squares solutions forced to belong to a vector semispace are also discussed. This constrained least-squares procedure furnishes a new theoretical basis to the foundation of QSAR or QSPR in the framework of quantum similarity and other approaches. (C) 2001 Elsevier Science B.V. All rights reserved.