# Geometric Matrix Midranges

## TL;DR

This paper introduces geometric matrix midranges for positive definite Hermitian matrices, exploring their properties, computational aspects, and extensions from two matrices to multiple matrices, with insights from linear algebra, geometry, and optimization.

## Contribution

It defines and analyzes geometric matrix midranges, extending the concept from two matrices to multiple matrices, and examines their properties from various mathematical perspectives.

## Key findings

- Comparison of matrix midrange with scalar and vector cases
- Significance of matrix midrange in computational contexts
- Analysis using linear algebra, differential geometry, and convex optimization

## Abstract

We define geometric matrix midranges for positive definite Hermitian matrices and study the midrange problem from a number of perspectives. Special attention is given to the midrange of two positive definite matrices before considering the extension of the problem to $N > 2$ matrices. We compare matrix midrange statistics with the scalar and vector midrange problem and note the special significance of the matrix problem from a computational standpoint. We also study various aspects of geometric matrix midrange statistics from the viewpoint of linear algebra, differential geometry and convex optimization.