The eigenvalue decomposition of normal matrices by the skew-symmetric part
This is an incremental improvement for numerical linear algebra, specifically targeting normal matrices with few real eigenvalues like random orthogonal matrices.
The paper tackles computing eigenvalue decompositions of dense real normal matrices by decomposing their skew-symmetric parts, showing that in most cases, the method matches the operation count of Hessenberg factorization.
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.