The eigenvalue decomposition of normal matrices by the skew-symmetric part

TL;DR

This paper introduces a novel eigenvalue decomposition method for dense real normal matrices using their skew-symmetric parts, offering efficiency and stability advantages especially for matrices with few real eigenvalues.

Contribution

The paper presents a new eigenvalue decomposition approach for normal matrices based on skew-symmetric decomposition, with analysis and practical experiments.

Findings

Method has similar operation count as Hessenberg factorization.

Performs well for matrices with few real eigenvalues.

Effective in computing Riemannian barycenters on SO(n).

Abstract

We propose a new method for computing the eigenvalue decomposition of a dense real normal matrix $A$ through the decomposition of its skew-symmetric part. The method relies on algorithms that are known to be efficiently implemented, such as the bidiagonal singular value decomposition and the symmetric eigenvalue decomposition. The advantages of this method stand for normal matrices with few real eigenvalues, such as random orthogonal matrices. We provide a stability and a complexity analysis of the method. The numerical performance is compared with existing algorithms. In most cases, the method has the same operation count as the Hessenberg factorization of a dense matrix. Finally, we provide experiments for the application of computing a Riemannian barycenter on the special orthogonal group.