Dynamical systems for eigenvalue problems of axisymmetric matrices with positive eigenvalues

TL;DR

This paper introduces a dynamical system called the S-Oja-Brockett equation for computing eigenvalues and eigenvectors of axisymmetric matrices with positive eigenvalues, demonstrating its global convergence.

Contribution

The paper proposes a novel dynamical system tailored for axisymmetric matrices that guarantees global convergence to eigenvalues and eigenvectors.

Findings

The S-Oja-Brockett equation converges globally to eigenvalues and eigenvectors.

The method is applicable to matrices with special axisymmetric structures.

Convergence properties are theoretically established.

Abstract

We consider the eigenvalues and eigenvectors of an axisymmetric matrix$A$ with some special structures. We propose S-Oja-Brockett equation $\frac{dX}{dt}=AXB-XBX^TSAX,$ where $X(t) \in {\mathbb R}^{n \times m}$ with $m \leq n$, $S$ is a positive definite symmetric solution of the Sylvester equation $A^TS = SA$ and $B$ is a real positive definite diagonal matrix whose diagonal elements are distinct each other, and show the S-Oja-Brockett equation has the global convergence to eigenvalues and its eigenvectors of $A$.