NormalEVP.jl

This repository contains routines for the eigenvalue decomposition of real normal matrices. It is associated with the paper :"Mataigne S., Gallivan K., The eigenvalue decomposition of normal matrices by the decomposition of the skew-symmetric part with applications to orthogonal matrices, (2024).

Requirements

To use this repository, the user must have a Julia installation and have installed the following packages for computations: LinearAlgebra, SkewLinearAlgebra. And also the following packages to make the plots: Plots, Colors, LaTeXStrings, Distributions, BenchmarkTools. These packages are easily obtained from the package installation environment as follows. In Julia REPL, press ] to access the installation environment and for each package, do

(@v1.6) pkg> add Name_of_Package

Use

• The folder src contains the files:
  • normal_schur.jl contains the main routines to compute the real Schur decomposition of a normal matrix, notably nrmschur(A::AbstractMatrix).
  • skewlog.jl contains the routine nrmskewlog(A::AbstractMatrix) to compute the matrix logarithm of an orthogonal matrix with determinant 1.
  • chase_zeros.jlcontains a routine to isolate the zero diagonal entries of an upper bidiagonal matrix, as described in the folder Note on zero chasing.
  • wxeigen.jl contains routines to obtain the eigenvalue decomposition of a symmetric matrix $A = [W\ -X; X\ W]$ where $W$ is symmetric and $X$ is skew-symmetric.
• The folder benchmark contains files to perform all experiments presented in the paper.
• The folder figurescontains all figures produced by the files from the benchmark folder.
• The folder Note on zero chasing briefly explains the routines from chase_zeros.jl.