Simultaneous hollowisation, joint numerical range, and stabilization by noise

TL;DR

This paper develops methods for transforming matrices into a form with equal diagonals, explores their relation to joint numerical ranges, and applies these techniques to stabilization problems using rotation and noise.

Contribution

It introduces new algorithms for simultaneous matrix transformations and links them to joint numerical ranges, with applications to stabilization by rotation and noise.

Findings

Efficient algorithms for simultaneous matrix transformations

Relation established between transformations and joint numerical range

Applications demonstrated in stabilization scenarios

Abstract

We consider orthogonal transformations of arbitrary square matrices to a form where all diagonal entries are equal. In our main results we treat the simultaneous transformation of two matrices and the symplectic orthogonal transformation of one matrix. A relation to the joint real numerical range is worked out, efficient numerical algorithms are developped and applications to stabilization by rotation and by noise are presented.