On positive definite preserving linear transformations of rank $r$ on real symmetric matrices

TL;DR

This paper investigates conditions under which certain linear transformations of real symmetric matrices preserve positive definiteness, providing necessary and sufficient conditions for low ranks and sufficient conditions for higher ranks.

Contribution

It characterizes positive definite preservation for linear transformations of rank 1 and 2, and offers sufficient conditions for higher ranks.

Findings

Necessary and sufficient conditions for rank 1 transformations.

Necessary and sufficient conditions for rank 2 transformations.

Sufficient conditions for transformations of rank r.

Abstract

We study on what conditions on $B_k,$ \ a linear transformation of rank $r$ \label{form} T(A)=\sum_{k=1}^r\tr(AB_k)U_k where $U_k,\ k=1,2,..., r$ are linear independent and all positive definite; is positive definite preserving. We give some first results for this question. For the case of rank one and two, the necessary and sufficient conditions are given. We also give some sufficient conditions for the case of rank $r.$