Positive definite preserving linear transformations on symmetric matrix spaces

TL;DR

This paper characterizes linear transformations on symmetric matrix spaces that preserve positive definiteness, under specific assumptions, using properties of the Hadamard product.

Contribution

It provides new characterizations of positive definite preserving linear transformations with additional assumptions on the transformation or the matrices.

Findings

Characterizations under the assumption T(E_{ii})=1 for all i

Results assuming T(A)>0 implies A>0

Insights into transformations preserving positive definiteness

Abstract

Base on some simple facts of Hadamard product, characterizations of positive definite preserving linear transformations on real symmetric matrix spaces with an additional assumption "$\ra T(E_{ii})=1, i=1,2,..., n$" or "$T(A)>0\to A> 0$", were given.