# Positive Semidefiniteness and Positive Definiteness of a Linear Parametric Interval Matrix

## TL;DR

This paper studies conditions for positive definiteness and semidefiniteness of symmetric matrices with linearly parameterized entries over interval domains, extending classical results and aiding optimization methods.

## Contribution

It provides new characterizations and computationally efficient conditions for positive (semi-)definiteness of parametric interval matrices, expanding classical theory.

## Key findings

- Characterization of positive definiteness over parameter intervals
- Efficient sufficient and necessary conditions proposed
- Extensions of classical interval matrix results

## Abstract

We consider a symmetric matrix, the entries of which depend linearly on some parameters. The domains of the parameters are compact real intervals. We investigate the problem of checking whether for each (or some) setting of the parameters, the matrix is positive definite (or positive semidefinite). We state a characterization in the form of equivalent conditions, and also propose some computationally cheap sufficient\,/\,necessary conditions. Our results extend the classical results on positive (semi-)definiteness of interval matrices. They may be useful for checking convexity or non-convexity in global optimization methods based on branch and bound framework and using interval techniques.