Successive Difference Substitution Based on Column Stochastic Matrix and Mechanical Decision for Positive Semi-definite Forms

TL;DR

This paper introduces a new successive difference substitution method based on column stochastic matrices to determine positive semi-definite forms, providing necessary and sufficient conditions and an automated Maple program for proof and counterexamples.

Contribution

It develops a novel SDS method using column stochastic matrices and establishes comprehensive criteria for positive semi-definiteness.

Findings

The method provides necessary and sufficient conditions for positive semi-definiteness.

The Maple program TSDS3 automates proof and counterexample generation.

The sequence of SDS sets for positive definite forms terminates positively.

Abstract

The theory part of this paper is sketched as follows. Based on column stochastic average matrix $T_n$ selected as a basic substitution matrix, the method of advanced successive difference substitution is established. Then, a set of necessary and sufficient conditions for deciding positive semi-definite form on $\R^n_+$ is derived from this method. And furthermore, it is proved that the sequence of SDS sets of a positive definite form is positively terminating. Worked out according to these results, the Maple program TSDS3 not only automatically proves the polynomial inequalities, but also outputs counter examples for the false. Sometimes TSDS3 does not halt, but it is very useful by experimenting on so many examples.