# Theory and Applications of Matrix-Weighted Consensus

## TL;DR

This paper introduces a matrix-weighted consensus algorithm that generalizes traditional scalar-weighted consensus, enabling new clustering and formation control applications in networked dynamical systems.

## Contribution

It develops a novel matrix-weighted consensus framework, providing algebraic conditions and algorithms for clustering and demonstrating practical applications.

## Key findings

- Consensus and clustering are achievable with matrix weights.
- The algorithm can identify all clusters in a system.
- Applications include clustered consensus and bearing-based formation control.

## Abstract

This paper proposes the matrix-weighted consensus algorithm, which is a generalization of the consensus algorithm in the literature. Given a networked dynamical system where the interconnections between agents are weighted by nonnegative definite matrices instead of nonnegative scalars, consensus and clustering phenomena naturally exist. We examine algebraic and algebraic graph conditions for achieving a consensus, and provide an algorithm for finding all clusters of a given system. Finally, we illustrate two applications of the proposed consensus algorithm in clustered consensus and in bearing-based formation control.