SkewLinearAlgebra

The SkewLinearAlgebra package provides specialized matrix types, optimized methods of LinearAlgebra functions, and a few entirely new functions for dealing with linear algebra on skew-Hermitian matrices, especially for the case of real skew-symmetric matrices.

Introduction

A skew-Hermitian matrix $A$ is a square matrix that equals the negative of its conjugate-transpose: $A=-\overline{A^{T}}=-A^{*}$, equivalent to A == -A' in Julia. (In the real skew-symmetric case, this is simply $A=-A^T$.) Such matrices have special computational properties: orthogonal eigenvectors and purely imaginary eigenvalues, "skew-Cholesky" factorizations, and a relative of the determinant called the Pfaffian.

Although any skew-Hermitian matrix $A$ can be transformed into a Hermitian matrix $H=iA$, this transformation converts real matrices $A$ into complex-Hermitian matrices $H$, which entails at least a factor of two loss in performance and memory usage compared to the real case. (And certain operations, like the Pfaffian, are only defined for the skew-symmetric case.) SkewLinearAlgebra gives you access to the greater performance and functionality that are possible for purely real skew-symmetric matrices.

To achieve this SkewLinearAlgebra defines a new matrix type, SkewHermitian (analogous to the LinearAlgebra.Hermitian type in the Julia standard library) that gives you access to optimized methods and specialized functionality for skew-Hermitian matrices, especially in the real case. It also provides a more specialized SkewHermTridiagonal for skew-Hermitian tridiagonal matrices (analogous to the LinearAlgebra.SymTridiagonal type in the Julia standard library) .

Contents

The SkewLinearAlgebra documentation is divided into the following sections:

• Skew-Hermitian matrices: the SkewHermitian type, constructors, and basic operations
• Skew-Hermitian tridiagonal matrices: the SkewHermTridiagonal type, constructors, and basic operations
• Skew-Hermitian eigenproblems: optimized eigenvalues/eigenvectors (& Schur/SVD), and Hessenberg factorization
• Trigonometric functions: optimized matrix exponentials and related functions (exp, sin, cos, etcetera)
• Pfaffian calculations: computation of the Pfaffian and log-Pfaffian
• Skew-Cholesky factorization: a skew-Hermitian analogue of Cholesky factorization

Quick start

Here is a simple example demonstrating some of the features of the SkewLinearAlgebra package. See the manual chapters outlines above for the complete details and explanations:

julia> using SkewLinearAlgebra, LinearAlgebra

julia> A = SkewHermitian([0  1 2
                         -1  0 3
                         -2 -3 0])
3×3 SkewHermitian{Int64, Matrix{Int64}}:
  0   1  2
 -1   0  3
 -2  -3  0

julia> eigvals(A) # optimized eigenvalue calculation (purely imaginary)
3-element Vector{ComplexF64}:
 0.0 - 3.7416573867739404im
 0.0 + 3.7416573867739404im
 0.0 + 0.0im

julia> Q = exp(A) # optimized matrix exponential
3×3 Matrix{Float64}:
  0.348107  -0.933192   0.0892929
 -0.63135   -0.303785  -0.713521
  0.692978   0.192007  -0.694921

julia> Q'Q ≈ I # the exponential of a skew-Hermitian matrix is unitary
true

julia> pfaffian(A) # the Pfaffian (always zero for odd-size skew matrices)
0.0

Acknowledgements

The SkewLinearAlgebra package was initially created by Simon Mataigne and Steven G. Johnson, with support from UCLouvain and the MIT–Belgium program.